{"history":[{"id":"2fac4e58-7d63-4470-b962-029c24b91553","wordChanged":"the","newWord":":::summary\n:::\n\n# \nIn mathematics, the **Riemann hypothesis** is the [conjecture](https://bliptext.com/articles/conjecture) that ilovemen [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) has its [zeros](https://bliptext.com/articles/root-of-a-function) only at the negative even integers and [complex numbers](https://bliptext.com/articles/complex-number) with [real part](https://bliptext.com/articles/real-part) 1⁄2.\nMany consider it to be the most important [unsolved problem](https://bliptext.com/articles/list-of-unsolved-problems-in-mathematics) in [pure mathematics](https://bliptext.com/articles/pure-mathematics).\nIt is of great interest in [number theory](https://bliptext.com/articles/number-theory) because it implies results about the distribution of [prime numbers](https://bliptext.com/articles/prime-numbers).\nIt was proposed by, after whom it is named.\nThe Riemann hypothesis and some of its generalizations, along with [Goldbach's conjecture](https://bliptext.com/articles/goldbach-s-conjecture) and the [twin prime conjecture](https://bliptext.com/articles/twin-prime-conjecture), make up [Hilbert's eighth problem](https://bliptext.com/articles/hilbert-s-eighth-problem) in [David Hilbert](https://bliptext.com/articles/david-hilbert)'s list of [twenty-three unsolved problems](https://bliptext.com/articles/hilbert-s-problems); it is also one of the [Millennium Prize Problems](https://bliptext.com/articles/millennium-prize-problems) of the [Clay Mathematics Institute](https://bliptext.com/articles/clay-mathematics-institute), which offers [US$](https://bliptext.com/articles/us)1 million for a solution to any of them.\nThe name is also used for some closely related analogues, such as the [Riemann hypothesis for curves over finite fields](https://bliptext.com/articles/riemann-hypothesis-for-curves-over-finite-fields).\nThe Riemann zeta [function](https://bliptext.com/articles/function-mathematics) ζ(s) is a [function](https://bliptext.com/articles/function-mathematics) whose [argument](https://bliptext.com/articles/argument-of-a-function) s may be any complex number other than 1, and whose values are also complex.\nIt has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros.\nThe zeta function is also zero for other values of s, which are called nontrivial zeros.\nThe Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:\nThus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers 1⁄2 + it, where t is a [real number](https://bliptext.com/articles/real-number) and i is the [imaginary unit](https://bliptext.com/articles/imaginary-unit).\n# [Riemann hypothesis | upload.wikimedia.org/wikipedia/commons/3/30/Riemann///zeta///function///absolute///value.png]\n\n# Riemann zeta function\nThe [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) is defined for complex s with real part greater than 1 by the [absolutely convergent](https://bliptext.com/articles/absolute-convergence) [infinite series](https://bliptext.com/articles/infinite-series)\n[Leonhard Euler](https://bliptext.com/articles/leonhard-euler) considered this series in the 1730s for real values of s, in conjunction with his solution to the [Basel problem](https://bliptext.com/articles/basel-problem).\nHe also proved that it equals the [Euler product](https://bliptext.com/articles/euler-product)\nwhere the [infinite product](https://bliptext.com/articles/infinite-product) extends over all prime numbers p.\nThe Riemann hypothesis discusses zeros outside the [region of convergence](https://bliptext.com/articles/region-of-convergence) of this series and Euler product.\nTo make sense of the hypothesis, it is necessary to [analytically continue](https://bliptext.com/articles/analytic-continuation) the function to obtain a form that is valid for all complex s.\nBecause the zeta function is [meromorphic](https://bliptext.com/articles/meromorphic), all choices of how to perform this analytic continuation will lead to the same result, by the [identity theorem](https://bliptext.com/articles/identity-theorem).\nA first step in this continuation observes that the series for the zeta function and the [Dirichlet eta function](https://bliptext.com/articles/dirichlet-eta-function) satisfy the relation\nwithin the region of convergence for both series.\nBut the eta function series on the right converges not just when the real part of s is greater than one, but more generally whenever s has positive real part.\nThus the zeta function can be redefined as, extending it from Re(s) > 1 to a larger domain: Re(s) > 0, except for the points where 1-2/2^s is zero.\nThese are the points where n can be any nonzero integer; the zeta function can be extended to these values too by taking limits (see ), giving a finite value for all values of s with positive real part except the [simple pole](https://bliptext.com/articles/simple-pole) at s = 1.\nIn the strip 0 < Re(s) < 1 this extension of the zeta function satisfies the [functional equation](https://bliptext.com/articles/riemann-zeta-function)\nOne may then define ζ(s) for all remaining nonzero complex numbers s (Re(s) ≤ 0 and s ≠ 0) by applying this equation outside the strip, and letting ζ(s) equal the right side of the equation whenever s has non-positive real part (and s ≠ 0).\nIf s is a negative even integer, then ζ(s) = 0, because the factor sin(πs/2) vanishes; these are the zeta function's trivial zeros.\n(If s is a positive even integer this argument does not apply because the zeros of the [sine](https://bliptext.com/articles/sine) function are canceled by the poles of the [gamma function](https://bliptext.com/articles/gamma-function) as it takes negative integer arguments.)\nThe value ζ(0) = −1/2 is not determined by the functional equation, but is the limiting value of ζ(s) as s approaches zero.\nThe functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros, so all nontrivial zeros lie in the critical strip where s has real part between 0 and 1.\n\n# Origin\n\"... es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien.\n... it is very probable that all roots are real.\nOf course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the immediate objective of my investigation.\" \"At the death of Riemann, a note was found among his papers, saying \"These properties of ζ(s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it.\"\nWe still have not the slightest idea of what the expression could be.\nAs to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself].\"\n- Jacques Hadamard\nRiemann's original motivation for studying the zeta function and its zeros was their occurrence in his [explicit formula](https://bliptext.com/articles/explicit-formulae-l-function) for the [number of primes](https://bliptext.com/articles/prime-counting-function) π(x) less than or equal to a given number x, which he published in his 1859 paper \"[On the Number of Primes Less Than a Given Magnitude](https://bliptext.com/articles/on-the-number-of-primes-less-than-a-given-magnitude)\".\nHis formula was given in terms of the related function\nwhich counts the primes and prime powers up to x, counting a prime power pn as 1/n.\nThe number of primes can be recovered from this function by using the [Möbius inversion formula](https://bliptext.com/articles/m-bius-inversion-formula),\nwhere μ is the [Möbius function](https://bliptext.com/articles/m-bius-function).\nRiemann's formula is then\nwhere the sum is over the nontrivial zeros of the zeta function and where Π0 is a slightly modified version of Π that replaces its value at its points of [discontinuity](https://bliptext.com/articles/discontinuity-mathematics) by the average of its upper and lower limits:\nThe summation in Riemann's formula is not absolutely convergent, but may be evaluated by taking the zeros ρ in order of the absolute value of their imaginary part.\nThe function li occurring in the first term is the (unoffset) [logarithmic integral function](https://bliptext.com/articles/logarithmic-integral-function) given by the [Cauchy principal value](https://bliptext.com/articles/cauchy-principal-value) of the divergent integral\nThe terms li(xρ) involving the zeros of the zeta function need some care in their definition as li has branch points at 0 and 1, and are defined (for x > 1) by analytic continuation in the complex variable ρ in the region Re(ρ) > 0, i.e. they should be considered as Ei(ρ log x).\nThe other terms also correspond to zeros: the dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros.\nFor some graphs of the sums of the first few terms of this series see or.\nThis formula says that the zeros of the Riemann zeta function control the [oscillations](https://bliptext.com/articles/oscillation) of primes around their \"expected\" positions.\nRiemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1.\nHe checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.\n\"The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely, number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes.\"\n\n# Consequences\nThe practical uses of the Riemann hypothesis include many propositions known to be true under the Riemann hypothesis, and some that can be shown to be equivalent to the Riemann hypothesis.\n\n## Distribution of prime numbers\n[Riemann's explicit formula](https://bliptext.com/articles/riemann-s-explicit-formula) for [the number of primes less than a given number](https://bliptext.com/articles/prime-counting-function) states that, in terms of a sum over the zeros of the Riemann zeta function, the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function.\nIn particular, the error term in the [prime number theorem](https://bliptext.com/articles/prime-number-theorem) is closely related to the position of the zeros.\nFor example, if β is the [upper bound](https://bliptext.com/articles/upper-bound) of the real parts of the zeros, then\n, where \\pi(x) is the [prime-counting function](https://bliptext.com/articles/prime-counting-function), is the [logarithmic integral function](https://bliptext.com/articles/logarithmic-integral-function), \\log(x) is the [natural logarithm](https://bliptext.com/articles/natural-logarithm) of x, and [big O notation](https://bliptext.com/articles/big-o-notation) is used here.\nIt is already known that 1/2 ≤ β ≤ 1.\n[Von Koch (1901)](https://bliptext.com/articles/) proved that the Riemann hypothesis implies the \"best possible\" bound for the error of the prime number theorem.\nA precise version of von Koch's result, due to, says that the Riemann hypothesis implies\nalso showed that the Riemann hypothesis implies\nwhere \\psi(x) is [Chebyshev's second function](https://bliptext.com/articles/chebyshev-function).\nproved that the Riemann hypothesis implies that for all x \\geq 2 there is a prime p satisfying\nThe constant 4/π may\nbe reduced to (1 + ε) provided that x is taken to be sufficiently large.\nThis is an explicit version of a theorem of [Cramér](https://bliptext.com/articles/cram-r).\n\n## Growth of arithmetic functions\nThe Riemann hypothesis implies strong bounds on the growth of many other [arithmetic functions](https://bliptext.com/articles/arithmetic-function), in addition to the primes counting function above.\nOne example involves the [Möbius function](https://bliptext.com/articles/m-bius-function) μ.\nThe statement that the equation\nis valid for every s with real part greater than 1/2, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis.\nFrom this we can also conclude that if the [Mertens function](https://bliptext.com/articles/mertens-function) is defined by\nthen the claim that\nfor every positive ε is equivalent to the Riemann hypothesis ([J. E. Littlewood](https://bliptext.com/articles/john-edensor-littlewood), 1912; see for instance: paragraph 14.25 in ).\nThe [determinant](https://bliptext.com/articles/determinant) of the order n [Redheffer matrix](https://bliptext.com/articles/redheffer-matrix) is equal to M(n), so the Riemann hypothesis can also be stated as a condition on the growth of these [determinant](https://bliptext.com/articles/determinant)s.\nLittlewood's result has been improved several times since then, by [Edmund Landau](https://bliptext.com/articles/edmund-landau), [Edward Charles Titchmarsh](https://bliptext.com/articles/edward-charles-titchmarsh), Helmut Maier and [Hugh Montgomery](https://bliptext.com/articles/hugh-lowell-montgomery), and [Kannan Soundararajan](https://bliptext.com/articles/kannan-soundararajan).\nSoundararajan's result is that, conditional on the Riemann hypothesis,\nThe Riemann hypothesis puts a rather tight bound on the growth of M, since disproved the slightly stronger [Mertens conjecture](https://bliptext.com/articles/mertens-conjecture)\nAnother closely related result is due to, that the Riemann hypothesis is equivalent to the statement that the [Euler characteristic](https://bliptext.com/articles/euler-characteristic) of the [simplicial complex](https://bliptext.com/articles/simplicial-complex) determined by the lattice of integers under divisibility is for all \\epsilon>0 (see [incidence algebra](https://bliptext.com/articles/incidence-algebra)).\nThe Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic functions aside from μ(n).\nA typical example is [Robin's theorem](https://bliptext.com/articles/robin-s-theorem), which states that if σ(n) is the [sigma function](https://bliptext.com/articles/divisor-function), given by\nthen\nfor all n > 5040 if and only if the Riemann hypothesis is true, where γ is the [Euler–Mascheroni constant](https://bliptext.com/articles/euler-mascheroni-constant).\nA related bound was given by [Jeffrey Lagarias](https://bliptext.com/articles/jeffrey-lagarias) in 2002, who proved that the Riemann hypothesis is equivalent to the statement that:\nfor every [natural number](https://bliptext.com/articles/natural-number) n > 1, where H_n is the nth [harmonic number](https://bliptext.com/articles/harmonic-number).\nThe Riemann hypothesis is also true if and only if the inequality\nis true for all n ≥ 120569# where φ(n) is [Euler's totient function](https://bliptext.com/articles/euler-s-totient-function) and 120569# is the [product of the first](https://bliptext.com/articles/primorial) 120569 primes.\nAnother example was found by [Jérôme Franel](https://bliptext.com/articles/j-r-me-franel), and extended by [Landau](https://bliptext.com/articles/edmund-landau) (see ).\nThe Riemann hypothesis is equivalent to several statements showing that the terms of the [Farey sequence](https://bliptext.com/articles/farey-sequence) are fairly regular.\nOne such equivalence is as follows: if Fn is the Farey sequence of order n, beginning with 1/n and up to 1/1, then the claim that for all ε > 0\nis equivalent to the Riemann hypothesis.\nHere\nis the number of terms in the Farey sequence of order n.\nFor an example from [group theory](https://bliptext.com/articles/group-theory), if g(n) is [Landau's function](https://bliptext.com/articles/landau-s-function) given by the maximal order of elements of the [symmetric group](https://bliptext.com/articles/symmetric-group) Sn of degree n, then showed that the Riemann hypothesis is equivalent to the bound\nfor all sufficiently large n.\n\n## Lindelöf hypothesis and growth of the zeta function\nThe Riemann hypothesis has various weaker consequences as well; one is the [Lindelöf hypothesis](https://bliptext.com/articles/lindel-f-hypothesis) on the rate of growth of the zeta function on the critical line, which says that, for any ε > 0,\nas.\nThe Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip.\nFor example, it implies that\nso the growth rate of ζ(1 + it) and its inverse would be known up to a factor of 2.\n\n## Large prime gap conjecture\nThe prime number theorem implies that on average, the [gap](https://bliptext.com/articles/prime-gap) between the prime p and its successor is log p.\nHowever, some gaps between primes may be much larger than the average.\n[Cramér](https://bliptext.com/articles/harald-cram-r) proved that, assuming the Riemann hypothesis, every gap is O(√p log p).\nThis is a case in which even the best bound that can be proved using the Riemann hypothesis is far weaker than what seems true: [Cramér's conjecture](https://bliptext.com/articles/cram-r-s-conjecture) implies that every gap is O((log p)2), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis.\nNumerical evidence supports Cramér's conjecture.\n\n## Analytic criteria equivalent to the Riemann hypothesis\nMany statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in proving (or disproving) it.\nSome typical examples are as follows.\n(Others involve the [divisor function](https://bliptext.com/articles/divisor-function) σ(n).)\nThe [Riesz criterion](https://bliptext.com/articles/riesz-criterion) was given by, to the effect that the bound\nholds for all ε > 0 if and only if the Riemann hypothesis holds.\nSee also the [Hardy–Littlewood criterion](https://bliptext.com/articles/riesz-function).\nproved that the Riemann hypothesis is true if and only if the space of functions of the form\nwhere ρ(z) is the fractional part of z, 0 ≤ θν ≤ 1, and\nis dense in the Hilbert space L2(0,1) of square-integrable functions on the unit interval.\nextended this by showing that the zeta function has no zeros with real part greater than 1/p if and only if this function space is dense in Lp(0,1).\nThis Nyman-Beurling criterion was strengthened by Baez-Duarte to the case where.\nshowed that the Riemann hypothesis is true if and only if the integral equation\nhas no non-trivial bounded solutions \\varphi for.\n[Weil's criterion](https://bliptext.com/articles/weil-s-criterion) is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis.\nRelated is [Li's criterion](https://bliptext.com/articles/li-s-criterion), a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis.\nproved that the Riemann hypothesis is equivalent to the statement that \\zeta'(s), the derivative of \\zeta(s), has no zeros in the strip\nThat \\zeta(s) has only simple zeros on the critical line is equivalent to its derivative having no zeros on the critical line.\nThe [Farey sequence](https://bliptext.com/articles/farey-sequence) provides two equivalences, due to [Jerome Franel](https://bliptext.com/articles/jerome-franel) and [Edmund Landau](https://bliptext.com/articles/edmund-landau) in 1924.\nThe [de Bruijn–Newman constant](https://bliptext.com/articles/de-bruijn-newman-constant) denoted by **Λ** and named after [Nicolaas Govert de Bruijn](https://bliptext.com/articles/nicolaas-govert-de-bruijn) and [Charles M. Newman](https://bliptext.com/articles/charles-m-newman), is defined\nas the unique real number such that the [function](https://bliptext.com/articles/function-mathematics)\nthat is parametrised by a [real](https://bliptext.com/articles/real-number) parameter λ, has a [complex](https://bliptext.com/articles/complex-number) variable z and is defined using a super-exponentially decaying function\nhas only real zeros if and only if λ ≥ Λ.\nSince the Riemann hypothesis is equivalent to the claim that all the zeroes of H(0, z) are real, the Riemann hypothesis is equivalent to the conjecture that.\nBrad Rodgers and [Terence Tao](https://bliptext.com/articles/terence-tao) discovered the equivalence is actually \\Lambda = 0 by proving zero to be the lower bound of the constant.\nProving zero is also the upper bound would therefore prove the Riemann hypothesis.\nAs of April 2020 the upper bound is.\n\n## Consequences of the generalized Riemann hypothesis\nSeveral applications use the [generalized Riemann hypothesis](https://bliptext.com/articles/generalized-riemann-hypothesis) for [Dirichlet L-series](https://bliptext.com/articles/dirichlet-l-series) or [zeta functions of number fields](https://bliptext.com/articles/dedekind-zeta-function) rather than just the Riemann hypothesis.\nMany basic properties of the Riemann zeta function can easily be generalized to all Dirichlet L-series, so it is plausible that a method that proves the Riemann hypothesis for the Riemann zeta function would also work for the generalized Riemann hypothesis for Dirichlet L-functions.\nSeveral results first proved using the generalized Riemann hypothesis were later given unconditional proofs without using it, though these were usually much harder.\nMany of the consequences on the following list are taken from.\n\n## Excluded middle\nSome consequences of the RH are also consequences of its negation, and are thus theorems.\nIn their discussion of the [Hecke, Deuring, Mordell, Heilbronn theorem](https://bliptext.com/articles/), say\n\"The method of proof here is truly amazing. If the generalized Riemann hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!!\"\nCare should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample.\n\n### Littlewood's theorem\nThis concerns the sign of the error in the [prime number theorem](https://bliptext.com/articles/prime-number-theorem).\nIt has been computed that π(x) < li(x) for all x ≤ 1025 (see this [table](https://bliptext.com/articles/prime-number-theorem)), and no value of x is known for which π(x) > li(x).\nIn 1914 Littlewood proved that there are arbitrarily large values of x for which\nand that there are also arbitrarily large values of x for which\nThus the difference π(x) − li(x) changes sign infinitely many times.\n[Skewes' number](https://bliptext.com/articles/skewes-number) is an estimate of the value of x corresponding to the first sign change.\nLittlewood's proof is divided into two cases: the RH is assumed false (about half a page of ), and the RH is assumed true (about a dozen pages).\nfollowed this up with a paper on the number of times \\Delta(n) changes sign in the interval \\Delta(n).\n\n### Gauss's class number conjecture\nThis is the [conjecture](https://bliptext.com/articles/class-number-problem) (first stated in article 303 of Gauss's [Disquisitiones Arithmeticae](https://bliptext.com/articles/disquisitiones-arithmeticae)) that there are only finitely many imaginary quadratic fields with a given class number.\nOne way to prove it would be to show that as the discriminant\nD → −∞\nthe class number\nh(D) → ∞\n.\nThe following sequence of theorems involving the Riemann hypothesis is described in :\n(In the work of Hecke and Heilbronn, the only L-functions that occur are those attached to imaginary quadratic characters, and it is only for those L-functions that GRH is true or GRH is false is intended; a failure of GRH for the L-function of a cubic Dirichlet character would, strictly speaking, mean GRH is false, but that was not the kind of failure of GRH that Heilbronn had in mind, so his assumption was more restricted than simply GRH is false.)\nIn 1935, [Carl Siegel](https://bliptext.com/articles/carl-siegel) strengthened the result without using RH or GRH in any way.\n\n### Growth of Euler's totient\nIn 1983 [J. L. Nicolas](https://bliptext.com/articles/jean-louis-nicolas) proved that\nfor infinitely many n, where φ(n) is [Euler's totient function](https://bliptext.com/articles/euler-s-totient-function) and γ is [Euler's constant](https://bliptext.com/articles/euler-s-constant).\nRibenboim remarks that: \"The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption.\"\n\n# Generalizations and analogs\n\n## Dirichlet L-series and other number fields\nThe Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, but much more general, global [L-functions](https://bliptext.com/articles/l-function).\nIn this broader setting, one expects the non-trivial zeros of the global L-functions to have real part 1/2.\nIt is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta function, which account for the true importance of the Riemann hypothesis in mathematics.\nThe [generalized Riemann hypothesis](https://bliptext.com/articles/generalized-riemann-hypothesis) extends the Riemann hypothesis to all [Dirichlet L-functions](https://bliptext.com/articles/dirichlet-l-function).\nIn particular it implies the conjecture that [Siegel zeros](https://bliptext.com/articles/siegel-zero) (zeros of L-functions between 1/2 and 1) do not exist.\nThe [extended Riemann hypothesis](https://bliptext.com/articles/extended-riemann-hypothesis) extends the Riemann hypothesis to all [Dedekind zeta functions](https://bliptext.com/articles/dedekind-zeta-function) of [algebraic number fields](https://bliptext.com/articles/algebraic-number-field).\nThe extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis.\nThe Riemann hypothesis can also be extended to the L-functions of [Hecke characters](https://bliptext.com/articles/hecke-character) of number fields.\nThe [grand Riemann hypothesis](https://bliptext.com/articles/grand-riemann-hypothesis) extends it to all [automorphic zeta functions](https://bliptext.com/articles/automorphic-l-function), such as [Mellin transforms](https://bliptext.com/articles/mellin-transform) of [Hecke eigenforms](https://bliptext.com/articles/hecke-eigenform).\n\n## Function fields and zeta functions of varieties over finite fields\nintroduced global zeta functions of (quadratic) [function fields](https://bliptext.com/articles/function-field-of-an-algebraic-variety) and conjectured an analogue of the Riemann hypothesis for them, which has been proved by Hasse in the genus 1 case and by in general.\nFor instance, the fact that the [Gauss sum](https://bliptext.com/articles/quadratic-gauss-sum), of the quadratic character of a [finite field](https://bliptext.com/articles/finite-field) of size q (with q odd), has absolute value \\sqrt{q} is actually an instance of the Riemann hypothesis in the function field setting.\nThis led to conjecture a similar statement for all [algebraic varieties](https://bliptext.com/articles/algebraic-variety); the resulting [Weil conjectures](https://bliptext.com/articles/weil-conjectures) were proved by.\n\n## Arithmetic zeta functions of arithmetic schemes and their L-factors\n[Arithmetic zeta functions](https://bliptext.com/articles/arithmetic-zeta-function) generalise the Riemann and Dedekind zeta functions as well as the zeta functions of varieties over finite fields to every arithmetic scheme or a scheme of finite type over integers.\nThe arithmetic zeta function of a regular connected [equidimensional](https://bliptext.com/articles/equidimensionality) arithmetic scheme of Kronecker dimension n can be factorized into the product of appropriately defined L-factors and an auxiliary factor.\nAssuming a functional equation and meromorphic continuation, the generalized Riemann hypothesis for the L-factor states that its zeros inside the critical strip lie on the central line.\nCorrespondingly, the generalized Riemann hypothesis for the arithmetic zeta function of a regular connected equidimensional arithmetic scheme states that its zeros inside the critical strip lie on vertical lines and its poles inside the critical strip lie on vertical lines.\nThis is known for schemes in positive characteristic and follows from, but remains entirely unknown in characteristic zero.\n\n## Selberg zeta functions\nintroduced the [Selberg zeta function](https://bliptext.com/articles/selberg-zeta-function) of a Riemann surface.\nThese are similar to the Riemann zeta function: they have a functional equation, and an infinite product similar to the Euler product but taken over closed geodesics rather than primes.\nThe [Selberg trace formula](https://bliptext.com/articles/selberg-trace-formula) is the analogue for these functions of the [explicit formulas](https://bliptext.com/articles/explicit-formula-l-function) in prime number theory.\nSelberg proved that the Selberg zeta functions satisfy the analogue of the Riemann hypothesis, with the imaginary parts of their zeros related to the [eigenvalues](https://bliptext.com/articles/eigenvalue) of the Laplacian operator of the Riemann surface.\n\n## Ihara zeta functions\nThe [Ihara zeta function](https://bliptext.com/articles/ihara-zeta-function) of a finite graph is an analogue of the [Selberg zeta function](https://bliptext.com/articles/selberg-zeta-function), which was first introduced by [Yasutaka Ihara](https://bliptext.com/articles/yasutaka-ihara) in the context of discrete subgroups of the two-by-two p-adic special linear group.\nA regular finite graph is a [Ramanujan graph](https://bliptext.com/articles/ramanujan-graph), a mathematical model of efficient communication networks, if and only if its Ihara zeta function satisfies the analogue of the Riemann hypothesis as was pointed out by [T. Sunada](https://bliptext.com/articles/toshikazu-sunada).\n\n## Montgomery's pair correlation conjecture\nsuggested the [pair correlation conjecture](https://bliptext.com/articles/pair-correlation-conjecture) that the correlation functions of the (suitably normalized) zeros of the zeta function should be the same as those of the eigenvalues of a [random hermitian matrix](https://bliptext.com/articles/gaussian-unitary-ensemble).\nshowed that this is supported by large-scale numerical calculations of these correlation functions.\nMontgomery showed that (assuming the Riemann hypothesis) at least 2/3 of all zeros are simple, and a related conjecture is that all zeros of the zeta function are simple (or more generally have no non-trivial integer linear relations between their imaginary parts).\n[Dedekind zeta functions](https://bliptext.com/articles/dedekind-zeta-function) of algebraic number fields, which generalize the Riemann zeta function, often do have multiple complex zeros.\nThis is because the Dedekind zeta functions factorize as a product of powers of [Artin L-functions](https://bliptext.com/articles/artin-l-function), so zeros of [Artin L-functions](https://bliptext.com/articles/artin-l-function) sometimes give rise to multiple zeros of Dedekind zeta functions.\nOther examples of zeta functions with multiple zeros are the L-functions of some [elliptic curves](https://bliptext.com/articles/elliptic-curve): these can have multiple zeros at the real point of their critical line; the [Birch-Swinnerton-Dyer conjecture](https://bliptext.com/articles/birch-swinnerton-dyer-conjecture) predicts that the multiplicity of this zero is the rank of the elliptic curve.\n\n## Other zeta functions\nThere are [many other examples](https://bliptext.com/articles/zeta-function) of zeta functions with analogues of the Riemann hypothesis, some of which have been proved.\n[Goss zeta functions](https://bliptext.com/articles/goss-zeta-function) of function fields have a Riemann hypothesis, proved by.\n[The main conjecture](https://bliptext.com/articles/main-conjecture-of-iwasawa-theory) of [Iwasawa theory](https://bliptext.com/articles/iwasawa-theory), proved by [Barry Mazur](https://bliptext.com/articles/barry-mazur) and [Andrew Wiles](https://bliptext.com/articles/andrew-wiles) for [cyclotomic fields](https://bliptext.com/articles/cyclotomic-field), and Wiles for [totally real fields](https://bliptext.com/articles/totally-real-number-field), identifies the zeros of a p-adic L-function with the eigenvalues of an operator, so can be thought of as an analogue of the [Hilbert–Pólya conjecture](https://bliptext.com/articles/hilbert-p-lya-conjecture) for p-adic L-functions.\n\n# Attempted proofs\nSeveral mathematicians have addressed the Riemann hypothesis, but none of their attempts has yet been accepted as a proof.\nlists some incorrect solutions.\n\n## Operator theory\nHilbert and Pólya suggested that one way to derive the Riemann hypothesis would be to find a [self-adjoint operator](https://bliptext.com/articles/self-adjoint-operator), from the existence of which the statement on the real parts of the zeros of ζ(s) would follow when one applies the criterion on real [eigenvalues](https://bliptext.com/articles/eigenvalue).\nSome support for this idea comes from several analogues of the Riemann zeta functions whose zeros correspond to eigenvalues of some operator: the zeros of a zeta function of a variety over a finite field correspond to eigenvalues of a [Frobenius element](https://bliptext.com/articles/frobenius-element) on an [étale cohomology](https://bliptext.com/articles/tale-cohomology) group, the zeros of a [Selberg zeta function](https://bliptext.com/articles/selberg-zeta-function) are eigenvalues of a [Laplacian operator](https://bliptext.com/articles/laplacian-operator) of a Riemann surface, and the zeros of a [p-adic zeta function](https://bliptext.com/articles/p-adic-zeta-function) correspond to eigenvectors of a Galois action on [ideal class groups](https://bliptext.com/articles/ideal-class-group).\nshowed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of [random matrices](https://bliptext.com/articles/random-matrices) drawn from the [Gaussian unitary ensemble](https://bliptext.com/articles/gaussian-unitary-ensemble).\nThis gives some support to the [Hilbert–Pólya conjecture](https://bliptext.com/articles/hilbert-p-lya-conjecture).\nIn 1999, [Michael Berry](https://bliptext.com/articles/michael-berry-physicist) and [Jonathan Keating](https://bliptext.com/articles/jonathan-keating) conjectured that there is some unknown quantization \\hat H of the classical Hamiltonian H = xp so that\nand even more strongly, that the Riemann zeros coincide with the spectrum of the operator.\nThis is in contrast to [canonical quantization](https://bliptext.com/articles/canonical-quantization), which leads to the [Heisenberg uncertainty principle](https://bliptext.com/articles/heisenberg-uncertainty-principle) and the [natural numbers](https://bliptext.com/articles/natural-numbers) as spectrum of the [quantum harmonic oscillator](https://bliptext.com/articles/quantum-harmonic-oscillator).\nThe crucial point is that the Hamiltonian should be a self-adjoint operator so that the quantization would be a realization of the Hilbert–Pólya program.\nIn a connection with this quantum mechanical problem Berry and Connes had proposed that the inverse of the potential of the Hamiltonian is connected to the [half-derivative](https://bliptext.com/articles/half-derivative) of the function\nthen, in Hilbert-Polya approach\nThis yields a Hamiltonian whose eigenvalues are the square of the imaginary part of the Riemann zeros, and also that the [functional determinant](https://bliptext.com/articles/functional-determinant) of this [Hamiltonian operator](https://bliptext.com/articles/hamiltonian-operator) is just the [Riemann Xi function](https://bliptext.com/articles/riemann-xi-function).\nIn fact the Riemann Xi function would be proportional to the functional determinant ([Hadamard product](https://bliptext.com/articles/hadamard-product-matrices))\nHowever this operator is not useful in practice since it includes the inverse function (implicit function ) of the potential but not the potential itself.\nThe analogy with the Riemann hypothesis over [finite fields](https://bliptext.com/articles/finite-field) suggests that the Hilbert space containing eigenvectors corresponding to the zeros might be some sort of first [cohomology group](https://bliptext.com/articles/cohomology-group) of the [spectrum](https://bliptext.com/articles/spectrum-of-a-ring) Spec (Z) of the integers.\ndescribed some of the attempts to find such a cohomology theory.\nconstructed a natural space of invariant functions on the upper half plane that has eigenvalues under the Laplacian operator that correspond to zeros of the Riemann zeta function—and remarked that in the unlikely event that one could show the existence of a suitable positive definite inner product on this space, the Riemann hypothesis would follow.\ndiscussed a related example, where due to a bizarre bug a computer program listed zeros of the Riemann zeta function as eigenvalues of the same [Laplacian operator](https://bliptext.com/articles/laplacian-operator).\nsurveyed some of the attempts to construct a suitable physical model related to the Riemann zeta function.\n\n## Lee–Yang theorem\nThe [Lee–Yang theorem](https://bliptext.com/articles/lee-yang-theorem) states that the zeros of certain [partition functions](https://bliptext.com/articles/partition-function-statistical-mechanics) in [statistical mechanics](https://bliptext.com/articles/statistical-mechanics) all lie on a \"critical line\" with their real part equal to 0, and this has led to some speculation about a relationship with the Riemann hypothesis.\n\n## Turán's result\nshowed that if the functions\nhave no zeros when the real part of s is greater than one then\nwhere λ(n) is the [Liouville function](https://bliptext.com/articles/liouville-function) given by (−1)r if n has r prime factors.\nHe showed that this in turn would imply that the Riemann hypothesis is true.\nBut proved that T(x) is negative for infinitely many x (and also disproved the closely related [Pólya conjecture](https://bliptext.com/articles/p-lya-conjecture)), and showed that the smallest such x is 72 185 376 951 205.\nshowed by numerical calculation that the finite [Dirichlet series](https://bliptext.com/articles/dirichlet-series) above for N=19 has a zero with real part greater than 1.\nTurán also showed that a somewhat weaker assumption, the nonexistence of zeros with real part greater than 1+N−1/2+ε for large N in the finite Dirichlet series above, would also imply the Riemann hypothesis, but showed that for all sufficiently large N these series have zeros with real part greater than 1 + (log log N)/(4 log N).\nTherefore, Turán's result is [vacuously true](https://bliptext.com/articles/vacuous-truth) and cannot help prove the Riemann hypothesis.\n\n## Noncommutative geometry\nhas described a relationship between the Riemann hypothesis and [noncommutative geometry](https://bliptext.com/articles/noncommutative-geometry), and showed that a suitable analog of the [Selberg trace formula](https://bliptext.com/articles/selberg-trace-formula) for the action of the [idèle class group](https://bliptext.com/articles/id-le-class-group) on the adèle class space would imply the Riemann hypothesis.\nSome of these ideas are elaborated in.\n\n## Hilbert spaces of entire functions\nshowed that the Riemann hypothesis would follow from a positivity condition on a certain [Hilbert space](https://bliptext.com/articles/hilbert-space) of [entire functions](https://bliptext.com/articles/entire-function).\nHowever showed that the necessary positivity conditions are not satisfied.\nDespite this obstacle, de Branges has continued to work on an attempted proof of the Riemann hypothesis along the same lines, but this has not been widely accepted by other mathematicians.\n\n## Quasicrystals\nThe Riemann hypothesis implies that the zeros of the zeta function form a [quasicrystal](https://bliptext.com/articles/quasicrystal), a distribution with discrete support whose [Fourier transform](https://bliptext.com/articles/fourier-transform) also has discrete support.\nsuggested trying to prove the Riemann hypothesis by classifying, or at least studying, 1-dimensional quasicrystals.\n\n## Arithmetic zeta functions of models of elliptic curves over number fields\nWhen one goes from geometric dimension one, e.g. an [algebraic number field](https://bliptext.com/articles/algebraic-number-field), to geometric dimension two, e.g. a regular model of an [elliptic curve](https://bliptext.com/articles/elliptic-curve) over a number field, the two-dimensional part of the generalized Riemann hypothesis for the [arithmetic zeta function](https://bliptext.com/articles/arithmetic-zeta-function) of the model deals with the poles of the zeta function.\nIn dimension one the study of the zeta integral in [Tate's thesis](https://bliptext.com/articles/tate-s-thesis) does not lead to new important information on the Riemann hypothesis.\nContrary to this, in dimension two work of [Ivan Fesenko](https://bliptext.com/articles/ivan-fesenko) on two-dimensional generalisation of Tate's thesis includes an integral representation of a zeta integral closely related to the zeta function.\nIn this new situation, not possible in dimension one, the poles of the zeta function can be studied via the zeta integral and associated adele groups.\nRelated conjecture of on the positivity of the fourth derivative of a boundary function associated to the zeta integral essentially implies the pole part of the generalized Riemann hypothesis.\nproved that the latter, together with some technical assumptions, implies Fesenko's conjecture.\n\n## Multiple zeta functions\nDeligne's proof of the Riemann hypothesis over finite fields used the zeta functions of product varieties, whose zeros and poles correspond to sums of zeros and poles of the original zeta function, in order to bound the real parts of the zeros of the original zeta function.\nBy analogy, introduced multiple zeta functions whose zeros and poles correspond to sums of zeros and poles of the Riemann zeta function.\nTo make the series converge he restricted to sums of zeros or poles all with non-negative imaginary part.\nSo far, the known bounds on the zeros and poles of the multiple zeta functions are not strong enough to give useful estimates for the zeros of the Riemann zeta function.\n\n# Location of the zeros\n\n## Number of zeros\nThe functional equation combined with the [argument principle](https://bliptext.com/articles/argument-principle) implies that the number of zeros of the zeta function with imaginary part between 0 and T is given by for s=1/2+iT, where the argument is defined by varying it continuously along the line with Im(s)=T, starting with argument 0 at ∞+iT.\nThis is the sum of a large but well understood term\nand a small but rather mysterious term\nSo the density of zeros with imaginary part near T is about log(T)/(2π), and the function S describes the small deviations from this.\nThe function S(t) jumps by 1 at each zero of the zeta function, and for t ≥ 8 it decreases [monotonically](https://bliptext.com/articles/monotonically-decreasing) between zeros with derivative close to −log t.\nproved that, if T > e, then\n[Karatsuba](https://bliptext.com/articles/anatolii-alexeevitch-karatsuba) (1996) proved that every interval (T, T+H] for contains at least points where the function S(t) changes sign.\nshowed that the average moments of even powers of S are given by\nThis suggests that S(T)/(log log T)1/2 resembles a [Gaussian random variable](https://bliptext.com/articles/gaussian-random-variable) with mean 0 and variance 2π2 ( proved this fact).\nIn particular |S(T)| is usually somewhere around (log log T)1/2, but occasionally much larger.\nThe exact order of growth of S(T) is not known.\nThere has been no unconditional improvement to Riemann's original bound S(T)=O(log T), though the Riemann hypothesis implies the slightly smaller bound S(T)=O(log T/log log T).\nThe true order of magnitude may be somewhat less than this, as random functions with the same distribution as S(T) tend to have growth of order about log(T)1/2.\nIn the other direction it cannot be too small: showed that S(T) ≠ o((log T)1/3/(log log T)7/3), and assuming the Riemann hypothesis Montgomery showed that S(T) ≠ o((log T)1/2/(log log T)1/2).\nNumerical calculations confirm that S grows very slowly: |S(T)| < 1 for T < 280, |S(T)| < 2 for T < 6 800 000, and the largest value of |S(T)| found so far is not much larger than 3.\nRiemann's estimate S(T) = O(log T) implies that the gaps between zeros are bounded, and Littlewood improved this slightly, showing that the gaps between their imaginary parts tend to 0.\n\n## Theorem of Hadamard and de la Vallée-Poussin\nand independently proved that no zeros could lie on the line Re(s) = 1.\nTogether with the functional equation and the fact that there are no zeros with real part greater than 1, this showed that all non-trivial zeros must lie in the interior of the critical strip 0 < Re(s) < 1.\nThis was a key step in their first proofs of the [prime number theorem](https://bliptext.com/articles/prime-number-theorem).\nBoth the original proofs that the zeta function has no zeros with real part 1 are similar, and depend on showing that if ζ(1 + it) vanishes, then ζ(1 + 2it) is singular, which is not possible.\nOne way of doing this is by using the inequality\nfor σ > 1, t real, and looking at the limit as σ → 1.\nThis inequality follows by taking the real part of the log of the Euler product to see that\nwhere the sum is over all prime powers pn, so that\nwhich is at least 1 because all the terms in the sum are positive, due to the inequality\n\n## Zero-free regions\nThe most extensive computer search by Platt and Trudgian for counter examples of the Riemann hypothesis has verified it for.\nBeyond that zero-free regions are known as inequalities concerning σ + it, which can be zeroes.\nThe oldest version is from [De la Vallée-Poussin (1899–1900)](https://bliptext.com/articles/), who proved there is a region without zeroes that satisfies 1 − σ ≥ C⁄log(t) for some positive constant C.\nIn other words, zeros cannot be too close to the line there is a zero-free region close to this line.\nThis has been enlarged by several authors using methods such as [Vinogradov's mean-value theorem](https://bliptext.com/articles/vinogradov-s-mean-value-theorem).\nThe most recent paper by Mossinghoff, Trudgian and Yang is from December 2022 and provides four zero-free regions that improved the previous results of Kevin Ford from 2002, Mossinghoff and Trudgian themselves from 2015 and Pace Nielsen's slight improvement of Ford from October 2022:\nThe paper also presents an improvement to the second zero-free region, whose bounds are unknown on account of |t| being merely assumed to be \"sufficiently large\" to fulfill the requirements of the paper's proof.\nThis region is\n.\n\n# Zeros on the critical line\nand showed there are infinitely many zeros on the critical line, by considering moments of certain functions related to the zeta function.\nproved that at least a (small) positive proportion of zeros lie on the line.\nimproved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, and improved this further to two-fifths.\nIn 2020, this estimate was extended to five-twelfths by Pratt, Robles, [Zaharescu](https://bliptext.com/articles/alexandru-zaharescu) and Zeindler by considering extended mollifiers that can accommodate higher order derivatives of the zeta function and their associated Kloosterman sums.\nMost zeros lie close to the critical line.\nMore precisely, showed that for any positive ε, the number of zeroes with real part at least 1/2+ε and imaginary part at between -T and T is O(T).\nCombined with the facts that zeroes on the critical strip are symmetric about the critical line and that the total number of zeroes in the critical strip is, [almost all](https://bliptext.com/articles/almost-all) non-trivial zeroes are within a distance ε of the critical line.\ngives several more precise versions of this result, called zero density estimates, which bound the number of zeros in regions with imaginary part at most T and real part at least 1/2+ε.\n\n## Hardy–Littlewood conjectures\nIn 1914 [Godfrey Harold Hardy](https://bliptext.com/articles/g-h-hardy) proved that has infinitely many real zeros.\nThe next two conjectures of [Hardy](https://bliptext.com/articles/g-h-hardy) and [John Edensor Littlewood](https://bliptext.com/articles/john-edensor-littlewood) on the distance between real zeros of and on the density of zeros of on the interval (T,T+H] for sufficiently large T > 0, and and with as small as possible value of a > 0, where is an arbitrarily small number, open two new directions in the investigation of the Riemann zeta function:\nLet N(T) be the total number of real zeros, and N_0(T) be the total number of zeros of odd order of the function lying on the interval (0,T]~.\n\n## Selberg's zeta function conjecture\ninvestigated the problem of Hardy–Littlewood 2 and proved that for any ε > 0 there exists such and c = c(ε) > 0, such that for T \\geq T_0 and the inequality is true.\nSelberg conjectured that this could be tightened to H=T^{0.5}.\nproved that for a fixed ε satisfying the condition 0 < ε < 0.001, a sufficiently large T and, , the interval (T, T+H) contains at least cH log(T) real zeros of the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) and therefore confirmed the Selberg conjecture.\nThe estimates of Selberg and Karatsuba can not be improved in respect of the order of growth as T → ∞.\nproved that an analog of the Selberg conjecture holds for almost all intervals (T, T+H],, where ε is an arbitrarily small fixed positive number. The Karatsuba method permits to investigate zeros of the Riemann zeta function on \"supershort\" intervals of the critical line, that is, on the intervals (T, T+H], the length H of which grows slower than any, even arbitrarily small degree T.\nIn particular, he proved that for any given numbers ε, satisfying the conditions almost all intervals (T, T+H] for contain at least zeros of the function . This estimate is quite close to the one that follows from the Riemann hypothesis.\n\n## Numerical calculations\nThe function\nhas the same zeros as the zeta function in the critical strip, and is real on the critical line because of the functional equation, so one can prove the existence of zeros exactly on the real line between two points by checking numerically that the function has opposite signs at these points.\nUsually one writes\nwhere Hardy's [Z function](https://bliptext.com/articles/z-function) and the [Riemann–Siegel theta function](https://bliptext.com/articles/riemann-siegel-theta-function) θ are uniquely defined by this and the condition that they are smooth real functions with θ(0) = 0.\nBy finding many intervals where the function Z changes sign one can show that there are many zeros on the critical line.\nTo verify the Riemann hypothesis up to a given [imaginary part](https://bliptext.com/articles/imaginary-part) T of the zeros, one also has to check that there are no further zeros off the line in this region.\nThis can be done by calculating the total number of zeros in the region using [Turing's method](https://bliptext.com/articles/turing-s-method) and checking that it is the same as the number of zeros found on the line.\nThis allows one to verify the Riemann hypothesis computationally up to any desired value of T (provided all the zeros of the zeta function in this region are simple and on the critical line).\nThese calculations can also be used to estimate \\pi(x) for finite ranges of x. For example, using the latest result from 2020 (zeros up to height ), it has been shown that\nIn general, this inequality holds if where T is the largest known value such that the Riemann hypothesis is true for all zeros \\rho with.\nSome calculations of zeros of the zeta function are listed below, where the \"height\" of a zero is the magnitude of its imaginary part, and the height of the nth zero is denoted by γn. So far all zeros that have been checked are on the critical line and are simple.\n(A multiple zero would cause problems for the zero finding algorithms, which depend on finding sign changes between zeros.) For tables of the zeros, see or.\n\n## Gram points\nA [Gram point](https://bliptext.com/articles/gram-point) is a point on the critical line 1/2 + it where the zeta function is real and non-zero.\nUsing the expression for the zeta function on the critical line, ζ(1/2 + it) = Z(t)e− iθ(t), where Hardy's function, Z, is real for real t, and θ is the [Riemann–Siegel theta function](https://bliptext.com/articles/riemann-siegel-theta-function), we see that zeta is real when sin(θ(t)) = 0.\nThis implies that θ(t) is an integer multiple of π, which allows for the location of Gram points to be calculated fairly easily by inverting the formula for θ.\nThey are usually numbered as gn for n = 0, 1, ..., where gn is the unique solution of θ(t) = nπ.\nGram observed that there was often exactly one zero of the zeta function between any two Gram points; Hutchinson called this observation **[Gram's law](https://bliptext.com/articles/gram-s-law)**.\nThere are several other closely related statements that are also sometimes called Gram's law: for example, (−1)nZ(gn) is usually positive, or Z(t) usually has opposite sign at consecutive Gram points.\nThe imaginary parts γn of the first few zeros (in blue) and the first few Gram points gn are given in the following table\nThe first failure of Gram's law occurs at the 127th zero and the Gram point g126, which are in the \"wrong\" order.\nA Gram point t is called good if the zeta function is positive at 1/2 + it.\nThe indices of the \"bad\" Gram points where Z has the \"wrong\" sign are 126, 134, 195, 211, ... . A Gram block is an interval bounded by two good Gram points such that all the Gram points between them are bad.\nA refinement of Gram's law called Rosser's rule due to says that Gram blocks often have the expected number of zeros in them (the same as the number of Gram intervals), even though some of the individual Gram intervals in the block may not have exactly one zero in them.\nFor example, the interval bounded by g125 and g127 is a Gram block containing a unique bad Gram point g126, and contains the expected number 2 of zeros although neither of its two Gram intervals contains a unique zero.\nRosser et al. checked that there were no exceptions to Rosser's rule in the first 3 million zeros, although there are infinitely many exceptions to Rosser's rule over the entire zeta function.\nGram's rule and Rosser's rule both say that in some sense zeros do not stray too far from their expected positions.\nThe distance of a zero from its expected position is controlled by the function S defined above, which grows extremely slowly: its average value is of the order of (log log T)1/2, which only reaches 2 for T around 1024.\nThis means that both rules hold most of the time for small T but eventually break down often.\nIndeed, showed that both Gram's law and Rosser's rule fail in a positive proportion of cases.\nTo be specific, it is expected that in about 66% one zero is enclosed by two successive Gram points, but in 17% no zero and in 17% two zeros are in such a Gram-interval on the long run.\n# [This is a polar plot of the first 20 real values rn of the zeta function along the critical line, ζ(1/2 + it), with t running from 0 to 50. The values of rn in this range are the first 10 non-trivial Riemann zeta function zeros and the first 10 Gram points, each labeled by n. Fifty red points have been plotted between each rn, and the zeros are projected onto concentric magenta rings scaled to show the relative distance between their values of t.\nGram's law states that the curve usually crosses the real axis once between zeros. | upload.wikimedia.org/wikipedia/commons/f/fc/RiemannZeta///Zeros.svg]\n\n# Arguments for and against the Riemann hypothesis\nMathematical papers about the Riemann hypothesis tend to be cautiously noncommittal about its truth.\nOf authors who express an opinion, most of them, such as and, imply that they expect (or at least hope) that it is true.\nThe few authors who express serious doubt about it include, who lists some reasons for skepticism, and , who flatly states that he believes it false, that there is no evidence for it and no imaginable reason it would be true.\nThe consensus of the survey articles (,, and ) is that the evidence for it is strong but not overwhelming, so that while it is probably true there is reasonable doubt.\nSome of the arguments for and against the Riemann hypothesis are listed by, , and , and include the following:\n\n## Popular expositions\nNote: Derbyshire 2003, Rockmore 2005, Sabbagh 2003a, Sabbagh 2003b, Sautoy 2003, and Watkins 2015 are non-technical.\nEdwards 1974, Patterson 1988, Borwein/Choi/Rooney/Weirathmueller 2008, Mazur/Stein 2015, Broughan 2017, and Nahin 2021 give mathematical introductions.\nTitchmarsh 1986, Ivić 1985, and Karatsuba/Voronin 1992 are advanced [monographs](https://bliptext.com/articles/monograph).","wordIndex":9,"createdAt":"2025-08-07T11:27:22.354Z","user":{"id":"7SQCRejConW9QVzC13OdaKuSacCK72XM","name":"triǳ","image":"https://cdn.discordapp.com/avatars/811496901282103297/040d3c3cf8926f55f70657defaafe5e8.png"}}],"article":{"id":"b6d63cec-0978-459b-b78a-600163c9d8a2","title":"Riemann hypothesis","slug":"riemann-hypothesis","content":":::summary\n:::\n\n# \nIn mathematics, the **Riemann hypothesis** is the [conjecture](https://bliptext.com/articles/conjecture) that ilovemen [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) has its [zeros](https://bliptext.com/articles/root-of-a-function) only at the negative even integers and [complex numbers](https://bliptext.com/articles/complex-number) with [real part](https://bliptext.com/articles/real-part) 1⁄2.\nMany consider it to be the most important [unsolved problem](https://bliptext.com/articles/list-of-unsolved-problems-in-mathematics) in [pure mathematics](https://bliptext.com/articles/pure-mathematics).\nIt is of great interest in [number theory](https://bliptext.com/articles/number-theory) because it implies results about the distribution of [prime numbers](https://bliptext.com/articles/prime-numbers).\nIt was proposed by, after whom it is named.\nThe Riemann hypothesis and some of its generalizations, along with [Goldbach's conjecture](https://bliptext.com/articles/goldbach-s-conjecture) and the [twin prime conjecture](https://bliptext.com/articles/twin-prime-conjecture), make up [Hilbert's eighth problem](https://bliptext.com/articles/hilbert-s-eighth-problem) in [David Hilbert](https://bliptext.com/articles/david-hilbert)'s list of [twenty-three unsolved problems](https://bliptext.com/articles/hilbert-s-problems); it is also one of the [Millennium Prize Problems](https://bliptext.com/articles/millennium-prize-problems) of the [Clay Mathematics Institute](https://bliptext.com/articles/clay-mathematics-institute), which offers [US$](https://bliptext.com/articles/us)1 million for a solution to any of them.\nThe name is also used for some closely related analogues, such as the [Riemann hypothesis for curves over finite fields](https://bliptext.com/articles/riemann-hypothesis-for-curves-over-finite-fields).\nThe Riemann zeta [function](https://bliptext.com/articles/function-mathematics) ζ(s) is a [function](https://bliptext.com/articles/function-mathematics) whose [argument](https://bliptext.com/articles/argument-of-a-function) s may be any complex number other than 1, and whose values are also complex.\nIt has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros.\nThe zeta function is also zero for other values of s, which are called nontrivial zeros.\nThe Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:\nThus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers 1⁄2 + it, where t is a [real number](https://bliptext.com/articles/real-number) and i is the [imaginary unit](https://bliptext.com/articles/imaginary-unit).\n# [Riemann hypothesis | upload.wikimedia.org/wikipedia/commons/3/30/Riemann///zeta///function///absolute///value.png]\n\n# Riemann zeta function\nThe [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) is defined for complex s with real part greater than 1 by the [absolutely convergent](https://bliptext.com/articles/absolute-convergence) [infinite series](https://bliptext.com/articles/infinite-series)\n[Leonhard Euler](https://bliptext.com/articles/leonhard-euler) considered this series in the 1730s for real values of s, in conjunction with his solution to the [Basel problem](https://bliptext.com/articles/basel-problem).\nHe also proved that it equals the [Euler product](https://bliptext.com/articles/euler-product)\nwhere the [infinite product](https://bliptext.com/articles/infinite-product) extends over all prime numbers p.\nThe Riemann hypothesis discusses zeros outside the [region of convergence](https://bliptext.com/articles/region-of-convergence) of this series and Euler product.\nTo make sense of the hypothesis, it is necessary to [analytically continue](https://bliptext.com/articles/analytic-continuation) the function to obtain a form that is valid for all complex s.\nBecause the zeta function is [meromorphic](https://bliptext.com/articles/meromorphic), all choices of how to perform this analytic continuation will lead to the same result, by the [identity theorem](https://bliptext.com/articles/identity-theorem).\nA first step in this continuation observes that the series for the zeta function and the [Dirichlet eta function](https://bliptext.com/articles/dirichlet-eta-function) satisfy the relation\nwithin the region of convergence for both series.\nBut the eta function series on the right converges not just when the real part of s is greater than one, but more generally whenever s has positive real part.\nThus the zeta function can be redefined as, extending it from Re(s) > 1 to a larger domain: Re(s) > 0, except for the points where 1-2/2^s is zero.\nThese are the points where n can be any nonzero integer; the zeta function can be extended to these values too by taking limits (see ), giving a finite value for all values of s with positive real part except the [simple pole](https://bliptext.com/articles/simple-pole) at s = 1.\nIn the strip 0 < Re(s) < 1 this extension of the zeta function satisfies the [functional equation](https://bliptext.com/articles/riemann-zeta-function)\nOne may then define ζ(s) for all remaining nonzero complex numbers s (Re(s) ≤ 0 and s ≠ 0) by applying this equation outside the strip, and letting ζ(s) equal the right side of the equation whenever s has non-positive real part (and s ≠ 0).\nIf s is a negative even integer, then ζ(s) = 0, because the factor sin(πs/2) vanishes; these are the zeta function's trivial zeros.\n(If s is a positive even integer this argument does not apply because the zeros of the [sine](https://bliptext.com/articles/sine) function are canceled by the poles of the [gamma function](https://bliptext.com/articles/gamma-function) as it takes negative integer arguments.)\nThe value ζ(0) = −1/2 is not determined by the functional equation, but is the limiting value of ζ(s) as s approaches zero.\nThe functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros, so all nontrivial zeros lie in the critical strip where s has real part between 0 and 1.\n\n# Origin\n\"... es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien.\n... it is very probable that all roots are real.\nOf course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the immediate objective of my investigation.\" \"At the death of Riemann, a note was found among his papers, saying \"These properties of ζ(s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it.\"\nWe still have not the slightest idea of what the expression could be.\nAs to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself].\"\n- Jacques Hadamard\nRiemann's original motivation for studying the zeta function and its zeros was their occurrence in his [explicit formula](https://bliptext.com/articles/explicit-formulae-l-function) for the [number of primes](https://bliptext.com/articles/prime-counting-function) π(x) less than or equal to a given number x, which he published in his 1859 paper \"[On the Number of Primes Less Than a Given Magnitude](https://bliptext.com/articles/on-the-number-of-primes-less-than-a-given-magnitude)\".\nHis formula was given in terms of the related function\nwhich counts the primes and prime powers up to x, counting a prime power pn as 1/n.\nThe number of primes can be recovered from this function by using the [Möbius inversion formula](https://bliptext.com/articles/m-bius-inversion-formula),\nwhere μ is the [Möbius function](https://bliptext.com/articles/m-bius-function).\nRiemann's formula is then\nwhere the sum is over the nontrivial zeros of the zeta function and where Π0 is a slightly modified version of Π that replaces its value at its points of [discontinuity](https://bliptext.com/articles/discontinuity-mathematics) by the average of its upper and lower limits:\nThe summation in Riemann's formula is not absolutely convergent, but may be evaluated by taking the zeros ρ in order of the absolute value of their imaginary part.\nThe function li occurring in the first term is the (unoffset) [logarithmic integral function](https://bliptext.com/articles/logarithmic-integral-function) given by the [Cauchy principal value](https://bliptext.com/articles/cauchy-principal-value) of the divergent integral\nThe terms li(xρ) involving the zeros of the zeta function need some care in their definition as li has branch points at 0 and 1, and are defined (for x > 1) by analytic continuation in the complex variable ρ in the region Re(ρ) > 0, i.e. they should be considered as Ei(ρ log x).\nThe other terms also correspond to zeros: the dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros.\nFor some graphs of the sums of the first few terms of this series see or.\nThis formula says that the zeros of the Riemann zeta function control the [oscillations](https://bliptext.com/articles/oscillation) of primes around their \"expected\" positions.\nRiemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1.\nHe checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.\n\"The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely, number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes.\"\n\n# Consequences\nThe practical uses of the Riemann hypothesis include many propositions known to be true under the Riemann hypothesis, and some that can be shown to be equivalent to the Riemann hypothesis.\n\n## Distribution of prime numbers\n[Riemann's explicit formula](https://bliptext.com/articles/riemann-s-explicit-formula) for [the number of primes less than a given number](https://bliptext.com/articles/prime-counting-function) states that, in terms of a sum over the zeros of the Riemann zeta function, the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function.\nIn particular, the error term in the [prime number theorem](https://bliptext.com/articles/prime-number-theorem) is closely related to the position of the zeros.\nFor example, if β is the [upper bound](https://bliptext.com/articles/upper-bound) of the real parts of the zeros, then\n, where \\pi(x) is the [prime-counting function](https://bliptext.com/articles/prime-counting-function), is the [logarithmic integral function](https://bliptext.com/articles/logarithmic-integral-function), \\log(x) is the [natural logarithm](https://bliptext.com/articles/natural-logarithm) of x, and [big O notation](https://bliptext.com/articles/big-o-notation) is used here.\nIt is already known that 1/2 ≤ β ≤ 1.\n[Von Koch (1901)](https://bliptext.com/articles/) proved that the Riemann hypothesis implies the \"best possible\" bound for the error of the prime number theorem.\nA precise version of von Koch's result, due to, says that the Riemann hypothesis implies\nalso showed that the Riemann hypothesis implies\nwhere \\psi(x) is [Chebyshev's second function](https://bliptext.com/articles/chebyshev-function).\nproved that the Riemann hypothesis implies that for all x \\geq 2 there is a prime p satisfying\nThe constant 4/π may\nbe reduced to (1 + ε) provided that x is taken to be sufficiently large.\nThis is an explicit version of a theorem of [Cramér](https://bliptext.com/articles/cram-r).\n\n## Growth of arithmetic functions\nThe Riemann hypothesis implies strong bounds on the growth of many other [arithmetic functions](https://bliptext.com/articles/arithmetic-function), in addition to the primes counting function above.\nOne example involves the [Möbius function](https://bliptext.com/articles/m-bius-function) μ.\nThe statement that the equation\nis valid for every s with real part greater than 1/2, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis.\nFrom this we can also conclude that if the [Mertens function](https://bliptext.com/articles/mertens-function) is defined by\nthen the claim that\nfor every positive ε is equivalent to the Riemann hypothesis ([J. E. Littlewood](https://bliptext.com/articles/john-edensor-littlewood), 1912; see for instance: paragraph 14.25 in ).\nThe [determinant](https://bliptext.com/articles/determinant) of the order n [Redheffer matrix](https://bliptext.com/articles/redheffer-matrix) is equal to M(n), so the Riemann hypothesis can also be stated as a condition on the growth of these [determinant](https://bliptext.com/articles/determinant)s.\nLittlewood's result has been improved several times since then, by [Edmund Landau](https://bliptext.com/articles/edmund-landau), [Edward Charles Titchmarsh](https://bliptext.com/articles/edward-charles-titchmarsh), Helmut Maier and [Hugh Montgomery](https://bliptext.com/articles/hugh-lowell-montgomery), and [Kannan Soundararajan](https://bliptext.com/articles/kannan-soundararajan).\nSoundararajan's result is that, conditional on the Riemann hypothesis,\nThe Riemann hypothesis puts a rather tight bound on the growth of M, since disproved the slightly stronger [Mertens conjecture](https://bliptext.com/articles/mertens-conjecture)\nAnother closely related result is due to, that the Riemann hypothesis is equivalent to the statement that the [Euler characteristic](https://bliptext.com/articles/euler-characteristic) of the [simplicial complex](https://bliptext.com/articles/simplicial-complex) determined by the lattice of integers under divisibility is for all \\epsilon>0 (see [incidence algebra](https://bliptext.com/articles/incidence-algebra)).\nThe Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic functions aside from μ(n).\nA typical example is [Robin's theorem](https://bliptext.com/articles/robin-s-theorem), which states that if σ(n) is the [sigma function](https://bliptext.com/articles/divisor-function), given by\nthen\nfor all n > 5040 if and only if the Riemann hypothesis is true, where γ is the [Euler–Mascheroni constant](https://bliptext.com/articles/euler-mascheroni-constant).\nA related bound was given by [Jeffrey Lagarias](https://bliptext.com/articles/jeffrey-lagarias) in 2002, who proved that the Riemann hypothesis is equivalent to the statement that:\nfor every [natural number](https://bliptext.com/articles/natural-number) n > 1, where H_n is the nth [harmonic number](https://bliptext.com/articles/harmonic-number).\nThe Riemann hypothesis is also true if and only if the inequality\nis true for all n ≥ 120569# where φ(n) is [Euler's totient function](https://bliptext.com/articles/euler-s-totient-function) and 120569# is the [product of the first](https://bliptext.com/articles/primorial) 120569 primes.\nAnother example was found by [Jérôme Franel](https://bliptext.com/articles/j-r-me-franel), and extended by [Landau](https://bliptext.com/articles/edmund-landau) (see ).\nThe Riemann hypothesis is equivalent to several statements showing that the terms of the [Farey sequence](https://bliptext.com/articles/farey-sequence) are fairly regular.\nOne such equivalence is as follows: if Fn is the Farey sequence of order n, beginning with 1/n and up to 1/1, then the claim that for all ε > 0\nis equivalent to the Riemann hypothesis.\nHere\nis the number of terms in the Farey sequence of order n.\nFor an example from [group theory](https://bliptext.com/articles/group-theory), if g(n) is [Landau's function](https://bliptext.com/articles/landau-s-function) given by the maximal order of elements of the [symmetric group](https://bliptext.com/articles/symmetric-group) Sn of degree n, then showed that the Riemann hypothesis is equivalent to the bound\nfor all sufficiently large n.\n\n## Lindelöf hypothesis and growth of the zeta function\nThe Riemann hypothesis has various weaker consequences as well; one is the [Lindelöf hypothesis](https://bliptext.com/articles/lindel-f-hypothesis) on the rate of growth of the zeta function on the critical line, which says that, for any ε > 0,\nas.\nThe Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip.\nFor example, it implies that\nso the growth rate of ζ(1 + it) and its inverse would be known up to a factor of 2.\n\n## Large prime gap conjecture\nThe prime number theorem implies that on average, the [gap](https://bliptext.com/articles/prime-gap) between the prime p and its successor is log p.\nHowever, some gaps between primes may be much larger than the average.\n[Cramér](https://bliptext.com/articles/harald-cram-r) proved that, assuming the Riemann hypothesis, every gap is O(√p log p).\nThis is a case in which even the best bound that can be proved using the Riemann hypothesis is far weaker than what seems true: [Cramér's conjecture](https://bliptext.com/articles/cram-r-s-conjecture) implies that every gap is O((log p)2), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis.\nNumerical evidence supports Cramér's conjecture.\n\n## Analytic criteria equivalent to the Riemann hypothesis\nMany statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in proving (or disproving) it.\nSome typical examples are as follows.\n(Others involve the [divisor function](https://bliptext.com/articles/divisor-function) σ(n).)\nThe [Riesz criterion](https://bliptext.com/articles/riesz-criterion) was given by, to the effect that the bound\nholds for all ε > 0 if and only if the Riemann hypothesis holds.\nSee also the [Hardy–Littlewood criterion](https://bliptext.com/articles/riesz-function).\nproved that the Riemann hypothesis is true if and only if the space of functions of the form\nwhere ρ(z) is the fractional part of z, 0 ≤ θν ≤ 1, and\nis dense in the Hilbert space L2(0,1) of square-integrable functions on the unit interval.\nextended this by showing that the zeta function has no zeros with real part greater than 1/p if and only if this function space is dense in Lp(0,1).\nThis Nyman-Beurling criterion was strengthened by Baez-Duarte to the case where.\nshowed that the Riemann hypothesis is true if and only if the integral equation\nhas no non-trivial bounded solutions \\varphi for.\n[Weil's criterion](https://bliptext.com/articles/weil-s-criterion) is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis.\nRelated is [Li's criterion](https://bliptext.com/articles/li-s-criterion), a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis.\nproved that the Riemann hypothesis is equivalent to the statement that \\zeta'(s), the derivative of \\zeta(s), has no zeros in the strip\nThat \\zeta(s) has only simple zeros on the critical line is equivalent to its derivative having no zeros on the critical line.\nThe [Farey sequence](https://bliptext.com/articles/farey-sequence) provides two equivalences, due to [Jerome Franel](https://bliptext.com/articles/jerome-franel) and [Edmund Landau](https://bliptext.com/articles/edmund-landau) in 1924.\nThe [de Bruijn–Newman constant](https://bliptext.com/articles/de-bruijn-newman-constant) denoted by **Λ** and named after [Nicolaas Govert de Bruijn](https://bliptext.com/articles/nicolaas-govert-de-bruijn) and [Charles M. Newman](https://bliptext.com/articles/charles-m-newman), is defined\nas the unique real number such that the [function](https://bliptext.com/articles/function-mathematics)\nthat is parametrised by a [real](https://bliptext.com/articles/real-number) parameter λ, has a [complex](https://bliptext.com/articles/complex-number) variable z and is defined using a super-exponentially decaying function\nhas only real zeros if and only if λ ≥ Λ.\nSince the Riemann hypothesis is equivalent to the claim that all the zeroes of H(0, z) are real, the Riemann hypothesis is equivalent to the conjecture that.\nBrad Rodgers and [Terence Tao](https://bliptext.com/articles/terence-tao) discovered the equivalence is actually \\Lambda = 0 by proving zero to be the lower bound of the constant.\nProving zero is also the upper bound would therefore prove the Riemann hypothesis.\nAs of April 2020 the upper bound is.\n\n## Consequences of the generalized Riemann hypothesis\nSeveral applications use the [generalized Riemann hypothesis](https://bliptext.com/articles/generalized-riemann-hypothesis) for [Dirichlet L-series](https://bliptext.com/articles/dirichlet-l-series) or [zeta functions of number fields](https://bliptext.com/articles/dedekind-zeta-function) rather than just the Riemann hypothesis.\nMany basic properties of the Riemann zeta function can easily be generalized to all Dirichlet L-series, so it is plausible that a method that proves the Riemann hypothesis for the Riemann zeta function would also work for the generalized Riemann hypothesis for Dirichlet L-functions.\nSeveral results first proved using the generalized Riemann hypothesis were later given unconditional proofs without using it, though these were usually much harder.\nMany of the consequences on the following list are taken from.\n\n## Excluded middle\nSome consequences of the RH are also consequences of its negation, and are thus theorems.\nIn their discussion of the [Hecke, Deuring, Mordell, Heilbronn theorem](https://bliptext.com/articles/), say\n\"The method of proof here is truly amazing. If the generalized Riemann hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!!\"\nCare should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample.\n\n### Littlewood's theorem\nThis concerns the sign of the error in the [prime number theorem](https://bliptext.com/articles/prime-number-theorem).\nIt has been computed that π(x) < li(x) for all x ≤ 1025 (see this [table](https://bliptext.com/articles/prime-number-theorem)), and no value of x is known for which π(x) > li(x).\nIn 1914 Littlewood proved that there are arbitrarily large values of x for which\nand that there are also arbitrarily large values of x for which\nThus the difference π(x) − li(x) changes sign infinitely many times.\n[Skewes' number](https://bliptext.com/articles/skewes-number) is an estimate of the value of x corresponding to the first sign change.\nLittlewood's proof is divided into two cases: the RH is assumed false (about half a page of ), and the RH is assumed true (about a dozen pages).\nfollowed this up with a paper on the number of times \\Delta(n) changes sign in the interval \\Delta(n).\n\n### Gauss's class number conjecture\nThis is the [conjecture](https://bliptext.com/articles/class-number-problem) (first stated in article 303 of Gauss's [Disquisitiones Arithmeticae](https://bliptext.com/articles/disquisitiones-arithmeticae)) that there are only finitely many imaginary quadratic fields with a given class number.\nOne way to prove it would be to show that as the discriminant\nD → −∞\nthe class number\nh(D) → ∞\n.\nThe following sequence of theorems involving the Riemann hypothesis is described in :\n(In the work of Hecke and Heilbronn, the only L-functions that occur are those attached to imaginary quadratic characters, and it is only for those L-functions that GRH is true or GRH is false is intended; a failure of GRH for the L-function of a cubic Dirichlet character would, strictly speaking, mean GRH is false, but that was not the kind of failure of GRH that Heilbronn had in mind, so his assumption was more restricted than simply GRH is false.)\nIn 1935, [Carl Siegel](https://bliptext.com/articles/carl-siegel) strengthened the result without using RH or GRH in any way.\n\n### Growth of Euler's totient\nIn 1983 [J. L. Nicolas](https://bliptext.com/articles/jean-louis-nicolas) proved that\nfor infinitely many n, where φ(n) is [Euler's totient function](https://bliptext.com/articles/euler-s-totient-function) and γ is [Euler's constant](https://bliptext.com/articles/euler-s-constant).\nRibenboim remarks that: \"The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption.\"\n\n# Generalizations and analogs\n\n## Dirichlet L-series and other number fields\nThe Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, but much more general, global [L-functions](https://bliptext.com/articles/l-function).\nIn this broader setting, one expects the non-trivial zeros of the global L-functions to have real part 1/2.\nIt is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta function, which account for the true importance of the Riemann hypothesis in mathematics.\nThe [generalized Riemann hypothesis](https://bliptext.com/articles/generalized-riemann-hypothesis) extends the Riemann hypothesis to all [Dirichlet L-functions](https://bliptext.com/articles/dirichlet-l-function).\nIn particular it implies the conjecture that [Siegel zeros](https://bliptext.com/articles/siegel-zero) (zeros of L-functions between 1/2 and 1) do not exist.\nThe [extended Riemann hypothesis](https://bliptext.com/articles/extended-riemann-hypothesis) extends the Riemann hypothesis to all [Dedekind zeta functions](https://bliptext.com/articles/dedekind-zeta-function) of [algebraic number fields](https://bliptext.com/articles/algebraic-number-field).\nThe extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis.\nThe Riemann hypothesis can also be extended to the L-functions of [Hecke characters](https://bliptext.com/articles/hecke-character) of number fields.\nThe [grand Riemann hypothesis](https://bliptext.com/articles/grand-riemann-hypothesis) extends it to all [automorphic zeta functions](https://bliptext.com/articles/automorphic-l-function), such as [Mellin transforms](https://bliptext.com/articles/mellin-transform) of [Hecke eigenforms](https://bliptext.com/articles/hecke-eigenform).\n\n## Function fields and zeta functions of varieties over finite fields\nintroduced global zeta functions of (quadratic) [function fields](https://bliptext.com/articles/function-field-of-an-algebraic-variety) and conjectured an analogue of the Riemann hypothesis for them, which has been proved by Hasse in the genus 1 case and by in general.\nFor instance, the fact that the [Gauss sum](https://bliptext.com/articles/quadratic-gauss-sum), of the quadratic character of a [finite field](https://bliptext.com/articles/finite-field) of size q (with q odd), has absolute value \\sqrt{q} is actually an instance of the Riemann hypothesis in the function field setting.\nThis led to conjecture a similar statement for all [algebraic varieties](https://bliptext.com/articles/algebraic-variety); the resulting [Weil conjectures](https://bliptext.com/articles/weil-conjectures) were proved by.\n\n## Arithmetic zeta functions of arithmetic schemes and their L-factors\n[Arithmetic zeta functions](https://bliptext.com/articles/arithmetic-zeta-function) generalise the Riemann and Dedekind zeta functions as well as the zeta functions of varieties over finite fields to every arithmetic scheme or a scheme of finite type over integers.\nThe arithmetic zeta function of a regular connected [equidimensional](https://bliptext.com/articles/equidimensionality) arithmetic scheme of Kronecker dimension n can be factorized into the product of appropriately defined L-factors and an auxiliary factor.\nAssuming a functional equation and meromorphic continuation, the generalized Riemann hypothesis for the L-factor states that its zeros inside the critical strip lie on the central line.\nCorrespondingly, the generalized Riemann hypothesis for the arithmetic zeta function of a regular connected equidimensional arithmetic scheme states that its zeros inside the critical strip lie on vertical lines and its poles inside the critical strip lie on vertical lines.\nThis is known for schemes in positive characteristic and follows from, but remains entirely unknown in characteristic zero.\n\n## Selberg zeta functions\nintroduced the [Selberg zeta function](https://bliptext.com/articles/selberg-zeta-function) of a Riemann surface.\nThese are similar to the Riemann zeta function: they have a functional equation, and an infinite product similar to the Euler product but taken over closed geodesics rather than primes.\nThe [Selberg trace formula](https://bliptext.com/articles/selberg-trace-formula) is the analogue for these functions of the [explicit formulas](https://bliptext.com/articles/explicit-formula-l-function) in prime number theory.\nSelberg proved that the Selberg zeta functions satisfy the analogue of the Riemann hypothesis, with the imaginary parts of their zeros related to the [eigenvalues](https://bliptext.com/articles/eigenvalue) of the Laplacian operator of the Riemann surface.\n\n## Ihara zeta functions\nThe [Ihara zeta function](https://bliptext.com/articles/ihara-zeta-function) of a finite graph is an analogue of the [Selberg zeta function](https://bliptext.com/articles/selberg-zeta-function), which was first introduced by [Yasutaka Ihara](https://bliptext.com/articles/yasutaka-ihara) in the context of discrete subgroups of the two-by-two p-adic special linear group.\nA regular finite graph is a [Ramanujan graph](https://bliptext.com/articles/ramanujan-graph), a mathematical model of efficient communication networks, if and only if its Ihara zeta function satisfies the analogue of the Riemann hypothesis as was pointed out by [T. Sunada](https://bliptext.com/articles/toshikazu-sunada).\n\n## Montgomery's pair correlation conjecture\nsuggested the [pair correlation conjecture](https://bliptext.com/articles/pair-correlation-conjecture) that the correlation functions of the (suitably normalized) zeros of the zeta function should be the same as those of the eigenvalues of a [random hermitian matrix](https://bliptext.com/articles/gaussian-unitary-ensemble).\nshowed that this is supported by large-scale numerical calculations of these correlation functions.\nMontgomery showed that (assuming the Riemann hypothesis) at least 2/3 of all zeros are simple, and a related conjecture is that all zeros of the zeta function are simple (or more generally have no non-trivial integer linear relations between their imaginary parts).\n[Dedekind zeta functions](https://bliptext.com/articles/dedekind-zeta-function) of algebraic number fields, which generalize the Riemann zeta function, often do have multiple complex zeros.\nThis is because the Dedekind zeta functions factorize as a product of powers of [Artin L-functions](https://bliptext.com/articles/artin-l-function), so zeros of [Artin L-functions](https://bliptext.com/articles/artin-l-function) sometimes give rise to multiple zeros of Dedekind zeta functions.\nOther examples of zeta functions with multiple zeros are the L-functions of some [elliptic curves](https://bliptext.com/articles/elliptic-curve): these can have multiple zeros at the real point of their critical line; the [Birch-Swinnerton-Dyer conjecture](https://bliptext.com/articles/birch-swinnerton-dyer-conjecture) predicts that the multiplicity of this zero is the rank of the elliptic curve.\n\n## Other zeta functions\nThere are [many other examples](https://bliptext.com/articles/zeta-function) of zeta functions with analogues of the Riemann hypothesis, some of which have been proved.\n[Goss zeta functions](https://bliptext.com/articles/goss-zeta-function) of function fields have a Riemann hypothesis, proved by.\n[The main conjecture](https://bliptext.com/articles/main-conjecture-of-iwasawa-theory) of [Iwasawa theory](https://bliptext.com/articles/iwasawa-theory), proved by [Barry Mazur](https://bliptext.com/articles/barry-mazur) and [Andrew Wiles](https://bliptext.com/articles/andrew-wiles) for [cyclotomic fields](https://bliptext.com/articles/cyclotomic-field), and Wiles for [totally real fields](https://bliptext.com/articles/totally-real-number-field), identifies the zeros of a p-adic L-function with the eigenvalues of an operator, so can be thought of as an analogue of the [Hilbert–Pólya conjecture](https://bliptext.com/articles/hilbert-p-lya-conjecture) for p-adic L-functions.\n\n# Attempted proofs\nSeveral mathematicians have addressed the Riemann hypothesis, but none of their attempts has yet been accepted as a proof.\nlists some incorrect solutions.\n\n## Operator theory\nHilbert and Pólya suggested that one way to derive the Riemann hypothesis would be to find a [self-adjoint operator](https://bliptext.com/articles/self-adjoint-operator), from the existence of which the statement on the real parts of the zeros of ζ(s) would follow when one applies the criterion on real [eigenvalues](https://bliptext.com/articles/eigenvalue).\nSome support for this idea comes from several analogues of the Riemann zeta functions whose zeros correspond to eigenvalues of some operator: the zeros of a zeta function of a variety over a finite field correspond to eigenvalues of a [Frobenius element](https://bliptext.com/articles/frobenius-element) on an [étale cohomology](https://bliptext.com/articles/tale-cohomology) group, the zeros of a [Selberg zeta function](https://bliptext.com/articles/selberg-zeta-function) are eigenvalues of a [Laplacian operator](https://bliptext.com/articles/laplacian-operator) of a Riemann surface, and the zeros of a [p-adic zeta function](https://bliptext.com/articles/p-adic-zeta-function) correspond to eigenvectors of a Galois action on [ideal class groups](https://bliptext.com/articles/ideal-class-group).\nshowed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of [random matrices](https://bliptext.com/articles/random-matrices) drawn from the [Gaussian unitary ensemble](https://bliptext.com/articles/gaussian-unitary-ensemble).\nThis gives some support to the [Hilbert–Pólya conjecture](https://bliptext.com/articles/hilbert-p-lya-conjecture).\nIn 1999, [Michael Berry](https://bliptext.com/articles/michael-berry-physicist) and [Jonathan Keating](https://bliptext.com/articles/jonathan-keating) conjectured that there is some unknown quantization \\hat H of the classical Hamiltonian H = xp so that\nand even more strongly, that the Riemann zeros coincide with the spectrum of the operator.\nThis is in contrast to [canonical quantization](https://bliptext.com/articles/canonical-quantization), which leads to the [Heisenberg uncertainty principle](https://bliptext.com/articles/heisenberg-uncertainty-principle) and the [natural numbers](https://bliptext.com/articles/natural-numbers) as spectrum of the [quantum harmonic oscillator](https://bliptext.com/articles/quantum-harmonic-oscillator).\nThe crucial point is that the Hamiltonian should be a self-adjoint operator so that the quantization would be a realization of the Hilbert–Pólya program.\nIn a connection with this quantum mechanical problem Berry and Connes had proposed that the inverse of the potential of the Hamiltonian is connected to the [half-derivative](https://bliptext.com/articles/half-derivative) of the function\nthen, in Hilbert-Polya approach\nThis yields a Hamiltonian whose eigenvalues are the square of the imaginary part of the Riemann zeros, and also that the [functional determinant](https://bliptext.com/articles/functional-determinant) of this [Hamiltonian operator](https://bliptext.com/articles/hamiltonian-operator) is just the [Riemann Xi function](https://bliptext.com/articles/riemann-xi-function).\nIn fact the Riemann Xi function would be proportional to the functional determinant ([Hadamard product](https://bliptext.com/articles/hadamard-product-matrices))\nHowever this operator is not useful in practice since it includes the inverse function (implicit function ) of the potential but not the potential itself.\nThe analogy with the Riemann hypothesis over [finite fields](https://bliptext.com/articles/finite-field) suggests that the Hilbert space containing eigenvectors corresponding to the zeros might be some sort of first [cohomology group](https://bliptext.com/articles/cohomology-group) of the [spectrum](https://bliptext.com/articles/spectrum-of-a-ring) Spec (Z) of the integers.\ndescribed some of the attempts to find such a cohomology theory.\nconstructed a natural space of invariant functions on the upper half plane that has eigenvalues under the Laplacian operator that correspond to zeros of the Riemann zeta function—and remarked that in the unlikely event that one could show the existence of a suitable positive definite inner product on this space, the Riemann hypothesis would follow.\ndiscussed a related example, where due to a bizarre bug a computer program listed zeros of the Riemann zeta function as eigenvalues of the same [Laplacian operator](https://bliptext.com/articles/laplacian-operator).\nsurveyed some of the attempts to construct a suitable physical model related to the Riemann zeta function.\n\n## Lee–Yang theorem\nThe [Lee–Yang theorem](https://bliptext.com/articles/lee-yang-theorem) states that the zeros of certain [partition functions](https://bliptext.com/articles/partition-function-statistical-mechanics) in [statistical mechanics](https://bliptext.com/articles/statistical-mechanics) all lie on a \"critical line\" with their real part equal to 0, and this has led to some speculation about a relationship with the Riemann hypothesis.\n\n## Turán's result\nshowed that if the functions\nhave no zeros when the real part of s is greater than one then\nwhere λ(n) is the [Liouville function](https://bliptext.com/articles/liouville-function) given by (−1)r if n has r prime factors.\nHe showed that this in turn would imply that the Riemann hypothesis is true.\nBut proved that T(x) is negative for infinitely many x (and also disproved the closely related [Pólya conjecture](https://bliptext.com/articles/p-lya-conjecture)), and showed that the smallest such x is 72 185 376 951 205.\nshowed by numerical calculation that the finite [Dirichlet series](https://bliptext.com/articles/dirichlet-series) above for N=19 has a zero with real part greater than 1.\nTurán also showed that a somewhat weaker assumption, the nonexistence of zeros with real part greater than 1+N−1/2+ε for large N in the finite Dirichlet series above, would also imply the Riemann hypothesis, but showed that for all sufficiently large N these series have zeros with real part greater than 1 + (log log N)/(4 log N).\nTherefore, Turán's result is [vacuously true](https://bliptext.com/articles/vacuous-truth) and cannot help prove the Riemann hypothesis.\n\n## Noncommutative geometry\nhas described a relationship between the Riemann hypothesis and [noncommutative geometry](https://bliptext.com/articles/noncommutative-geometry), and showed that a suitable analog of the [Selberg trace formula](https://bliptext.com/articles/selberg-trace-formula) for the action of the [idèle class group](https://bliptext.com/articles/id-le-class-group) on the adèle class space would imply the Riemann hypothesis.\nSome of these ideas are elaborated in.\n\n## Hilbert spaces of entire functions\nshowed that the Riemann hypothesis would follow from a positivity condition on a certain [Hilbert space](https://bliptext.com/articles/hilbert-space) of [entire functions](https://bliptext.com/articles/entire-function).\nHowever showed that the necessary positivity conditions are not satisfied.\nDespite this obstacle, de Branges has continued to work on an attempted proof of the Riemann hypothesis along the same lines, but this has not been widely accepted by other mathematicians.\n\n## Quasicrystals\nThe Riemann hypothesis implies that the zeros of the zeta function form a [quasicrystal](https://bliptext.com/articles/quasicrystal), a distribution with discrete support whose [Fourier transform](https://bliptext.com/articles/fourier-transform) also has discrete support.\nsuggested trying to prove the Riemann hypothesis by classifying, or at least studying, 1-dimensional quasicrystals.\n\n## Arithmetic zeta functions of models of elliptic curves over number fields\nWhen one goes from geometric dimension one, e.g. an [algebraic number field](https://bliptext.com/articles/algebraic-number-field), to geometric dimension two, e.g. a regular model of an [elliptic curve](https://bliptext.com/articles/elliptic-curve) over a number field, the two-dimensional part of the generalized Riemann hypothesis for the [arithmetic zeta function](https://bliptext.com/articles/arithmetic-zeta-function) of the model deals with the poles of the zeta function.\nIn dimension one the study of the zeta integral in [Tate's thesis](https://bliptext.com/articles/tate-s-thesis) does not lead to new important information on the Riemann hypothesis.\nContrary to this, in dimension two work of [Ivan Fesenko](https://bliptext.com/articles/ivan-fesenko) on two-dimensional generalisation of Tate's thesis includes an integral representation of a zeta integral closely related to the zeta function.\nIn this new situation, not possible in dimension one, the poles of the zeta function can be studied via the zeta integral and associated adele groups.\nRelated conjecture of on the positivity of the fourth derivative of a boundary function associated to the zeta integral essentially implies the pole part of the generalized Riemann hypothesis.\nproved that the latter, together with some technical assumptions, implies Fesenko's conjecture.\n\n## Multiple zeta functions\nDeligne's proof of the Riemann hypothesis over finite fields used the zeta functions of product varieties, whose zeros and poles correspond to sums of zeros and poles of the original zeta function, in order to bound the real parts of the zeros of the original zeta function.\nBy analogy, introduced multiple zeta functions whose zeros and poles correspond to sums of zeros and poles of the Riemann zeta function.\nTo make the series converge he restricted to sums of zeros or poles all with non-negative imaginary part.\nSo far, the known bounds on the zeros and poles of the multiple zeta functions are not strong enough to give useful estimates for the zeros of the Riemann zeta function.\n\n# Location of the zeros\n\n## Number of zeros\nThe functional equation combined with the [argument principle](https://bliptext.com/articles/argument-principle) implies that the number of zeros of the zeta function with imaginary part between 0 and T is given by for s=1/2+iT, where the argument is defined by varying it continuously along the line with Im(s)=T, starting with argument 0 at ∞+iT.\nThis is the sum of a large but well understood term\nand a small but rather mysterious term\nSo the density of zeros with imaginary part near T is about log(T)/(2π), and the function S describes the small deviations from this.\nThe function S(t) jumps by 1 at each zero of the zeta function, and for t ≥ 8 it decreases [monotonically](https://bliptext.com/articles/monotonically-decreasing) between zeros with derivative close to −log t.\nproved that, if T > e, then\n[Karatsuba](https://bliptext.com/articles/anatolii-alexeevitch-karatsuba) (1996) proved that every interval (T, T+H] for contains at least points where the function S(t) changes sign.\nshowed that the average moments of even powers of S are given by\nThis suggests that S(T)/(log log T)1/2 resembles a [Gaussian random variable](https://bliptext.com/articles/gaussian-random-variable) with mean 0 and variance 2π2 ( proved this fact).\nIn particular |S(T)| is usually somewhere around (log log T)1/2, but occasionally much larger.\nThe exact order of growth of S(T) is not known.\nThere has been no unconditional improvement to Riemann's original bound S(T)=O(log T), though the Riemann hypothesis implies the slightly smaller bound S(T)=O(log T/log log T).\nThe true order of magnitude may be somewhat less than this, as random functions with the same distribution as S(T) tend to have growth of order about log(T)1/2.\nIn the other direction it cannot be too small: showed that S(T) ≠ o((log T)1/3/(log log T)7/3), and assuming the Riemann hypothesis Montgomery showed that S(T) ≠ o((log T)1/2/(log log T)1/2).\nNumerical calculations confirm that S grows very slowly: |S(T)| < 1 for T < 280, |S(T)| < 2 for T < 6 800 000, and the largest value of |S(T)| found so far is not much larger than 3.\nRiemann's estimate S(T) = O(log T) implies that the gaps between zeros are bounded, and Littlewood improved this slightly, showing that the gaps between their imaginary parts tend to 0.\n\n## Theorem of Hadamard and de la Vallée-Poussin\nand independently proved that no zeros could lie on the line Re(s) = 1.\nTogether with the functional equation and the fact that there are no zeros with real part greater than 1, this showed that all non-trivial zeros must lie in the interior of the critical strip 0 < Re(s) < 1.\nThis was a key step in their first proofs of the [prime number theorem](https://bliptext.com/articles/prime-number-theorem).\nBoth the original proofs that the zeta function has no zeros with real part 1 are similar, and depend on showing that if ζ(1 + it) vanishes, then ζ(1 + 2it) is singular, which is not possible.\nOne way of doing this is by using the inequality\nfor σ > 1, t real, and looking at the limit as σ → 1.\nThis inequality follows by taking the real part of the log of the Euler product to see that\nwhere the sum is over all prime powers pn, so that\nwhich is at least 1 because all the terms in the sum are positive, due to the inequality\n\n## Zero-free regions\nThe most extensive computer search by Platt and Trudgian for counter examples of the Riemann hypothesis has verified it for.\nBeyond that zero-free regions are known as inequalities concerning σ + it, which can be zeroes.\nThe oldest version is from [De la Vallée-Poussin (1899–1900)](https://bliptext.com/articles/), who proved there is a region without zeroes that satisfies 1 − σ ≥ C⁄log(t) for some positive constant C.\nIn other words, zeros cannot be too close to the line there is a zero-free region close to this line.\nThis has been enlarged by several authors using methods such as [Vinogradov's mean-value theorem](https://bliptext.com/articles/vinogradov-s-mean-value-theorem).\nThe most recent paper by Mossinghoff, Trudgian and Yang is from December 2022 and provides four zero-free regions that improved the previous results of Kevin Ford from 2002, Mossinghoff and Trudgian themselves from 2015 and Pace Nielsen's slight improvement of Ford from October 2022:\nThe paper also presents an improvement to the second zero-free region, whose bounds are unknown on account of |t| being merely assumed to be \"sufficiently large\" to fulfill the requirements of the paper's proof.\nThis region is\n.\n\n# Zeros on the critical line\nand showed there are infinitely many zeros on the critical line, by considering moments of certain functions related to the zeta function.\nproved that at least a (small) positive proportion of zeros lie on the line.\nimproved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, and improved this further to two-fifths.\nIn 2020, this estimate was extended to five-twelfths by Pratt, Robles, [Zaharescu](https://bliptext.com/articles/alexandru-zaharescu) and Zeindler by considering extended mollifiers that can accommodate higher order derivatives of the zeta function and their associated Kloosterman sums.\nMost zeros lie close to the critical line.\nMore precisely, showed that for any positive ε, the number of zeroes with real part at least 1/2+ε and imaginary part at between -T and T is O(T).\nCombined with the facts that zeroes on the critical strip are symmetric about the critical line and that the total number of zeroes in the critical strip is, [almost all](https://bliptext.com/articles/almost-all) non-trivial zeroes are within a distance ε of the critical line.\ngives several more precise versions of this result, called zero density estimates, which bound the number of zeros in regions with imaginary part at most T and real part at least 1/2+ε.\n\n## Hardy–Littlewood conjectures\nIn 1914 [Godfrey Harold Hardy](https://bliptext.com/articles/g-h-hardy) proved that has infinitely many real zeros.\nThe next two conjectures of [Hardy](https://bliptext.com/articles/g-h-hardy) and [John Edensor Littlewood](https://bliptext.com/articles/john-edensor-littlewood) on the distance between real zeros of and on the density of zeros of on the interval (T,T+H] for sufficiently large T > 0, and and with as small as possible value of a > 0, where is an arbitrarily small number, open two new directions in the investigation of the Riemann zeta function:\nLet N(T) be the total number of real zeros, and N_0(T) be the total number of zeros of odd order of the function lying on the interval (0,T]~.\n\n## Selberg's zeta function conjecture\ninvestigated the problem of Hardy–Littlewood 2 and proved that for any ε > 0 there exists such and c = c(ε) > 0, such that for T \\geq T_0 and the inequality is true.\nSelberg conjectured that this could be tightened to H=T^{0.5}.\nproved that for a fixed ε satisfying the condition 0 < ε < 0.001, a sufficiently large T and, , the interval (T, T+H) contains at least cH log(T) real zeros of the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) and therefore confirmed the Selberg conjecture.\nThe estimates of Selberg and Karatsuba can not be improved in respect of the order of growth as T → ∞.\nproved that an analog of the Selberg conjecture holds for almost all intervals (T, T+H],, where ε is an arbitrarily small fixed positive number. The Karatsuba method permits to investigate zeros of the Riemann zeta function on \"supershort\" intervals of the critical line, that is, on the intervals (T, T+H], the length H of which grows slower than any, even arbitrarily small degree T.\nIn particular, he proved that for any given numbers ε, satisfying the conditions almost all intervals (T, T+H] for contain at least zeros of the function . This estimate is quite close to the one that follows from the Riemann hypothesis.\n\n## Numerical calculations\nThe function\nhas the same zeros as the zeta function in the critical strip, and is real on the critical line because of the functional equation, so one can prove the existence of zeros exactly on the real line between two points by checking numerically that the function has opposite signs at these points.\nUsually one writes\nwhere Hardy's [Z function](https://bliptext.com/articles/z-function) and the [Riemann–Siegel theta function](https://bliptext.com/articles/riemann-siegel-theta-function) θ are uniquely defined by this and the condition that they are smooth real functions with θ(0) = 0.\nBy finding many intervals where the function Z changes sign one can show that there are many zeros on the critical line.\nTo verify the Riemann hypothesis up to a given [imaginary part](https://bliptext.com/articles/imaginary-part) T of the zeros, one also has to check that there are no further zeros off the line in this region.\nThis can be done by calculating the total number of zeros in the region using [Turing's method](https://bliptext.com/articles/turing-s-method) and checking that it is the same as the number of zeros found on the line.\nThis allows one to verify the Riemann hypothesis computationally up to any desired value of T (provided all the zeros of the zeta function in this region are simple and on the critical line).\nThese calculations can also be used to estimate \\pi(x) for finite ranges of x. For example, using the latest result from 2020 (zeros up to height ), it has been shown that\nIn general, this inequality holds if where T is the largest known value such that the Riemann hypothesis is true for all zeros \\rho with.\nSome calculations of zeros of the zeta function are listed below, where the \"height\" of a zero is the magnitude of its imaginary part, and the height of the nth zero is denoted by γn. So far all zeros that have been checked are on the critical line and are simple.\n(A multiple zero would cause problems for the zero finding algorithms, which depend on finding sign changes between zeros.) For tables of the zeros, see or.\n\n## Gram points\nA [Gram point](https://bliptext.com/articles/gram-point) is a point on the critical line 1/2 + it where the zeta function is real and non-zero.\nUsing the expression for the zeta function on the critical line, ζ(1/2 + it) = Z(t)e− iθ(t), where Hardy's function, Z, is real for real t, and θ is the [Riemann–Siegel theta function](https://bliptext.com/articles/riemann-siegel-theta-function), we see that zeta is real when sin(θ(t)) = 0.\nThis implies that θ(t) is an integer multiple of π, which allows for the location of Gram points to be calculated fairly easily by inverting the formula for θ.\nThey are usually numbered as gn for n = 0, 1, ..., where gn is the unique solution of θ(t) = nπ.\nGram observed that there was often exactly one zero of the zeta function between any two Gram points; Hutchinson called this observation **[Gram's law](https://bliptext.com/articles/gram-s-law)**.\nThere are several other closely related statements that are also sometimes called Gram's law: for example, (−1)nZ(gn) is usually positive, or Z(t) usually has opposite sign at consecutive Gram points.\nThe imaginary parts γn of the first few zeros (in blue) and the first few Gram points gn are given in the following table\nThe first failure of Gram's law occurs at the 127th zero and the Gram point g126, which are in the \"wrong\" order.\nA Gram point t is called good if the zeta function is positive at 1/2 + it.\nThe indices of the \"bad\" Gram points where Z has the \"wrong\" sign are 126, 134, 195, 211, ... . A Gram block is an interval bounded by two good Gram points such that all the Gram points between them are bad.\nA refinement of Gram's law called Rosser's rule due to says that Gram blocks often have the expected number of zeros in them (the same as the number of Gram intervals), even though some of the individual Gram intervals in the block may not have exactly one zero in them.\nFor example, the interval bounded by g125 and g127 is a Gram block containing a unique bad Gram point g126, and contains the expected number 2 of zeros although neither of its two Gram intervals contains a unique zero.\nRosser et al. checked that there were no exceptions to Rosser's rule in the first 3 million zeros, although there are infinitely many exceptions to Rosser's rule over the entire zeta function.\nGram's rule and Rosser's rule both say that in some sense zeros do not stray too far from their expected positions.\nThe distance of a zero from its expected position is controlled by the function S defined above, which grows extremely slowly: its average value is of the order of (log log T)1/2, which only reaches 2 for T around 1024.\nThis means that both rules hold most of the time for small T but eventually break down often.\nIndeed, showed that both Gram's law and Rosser's rule fail in a positive proportion of cases.\nTo be specific, it is expected that in about 66% one zero is enclosed by two successive Gram points, but in 17% no zero and in 17% two zeros are in such a Gram-interval on the long run.\n# [This is a polar plot of the first 20 real values rn of the zeta function along the critical line, ζ(1/2 + it), with t running from 0 to 50. The values of rn in this range are the first 10 non-trivial Riemann zeta function zeros and the first 10 Gram points, each labeled by n. Fifty red points have been plotted between each rn, and the zeros are projected onto concentric magenta rings scaled to show the relative distance between their values of t.\nGram's law states that the curve usually crosses the real axis once between zeros. | upload.wikimedia.org/wikipedia/commons/f/fc/RiemannZeta///Zeros.svg]\n\n# Arguments for and against the Riemann hypothesis\nMathematical papers about the Riemann hypothesis tend to be cautiously noncommittal about its truth.\nOf authors who express an opinion, most of them, such as and, imply that they expect (or at least hope) that it is true.\nThe few authors who express serious doubt about it include, who lists some reasons for skepticism, and , who flatly states that he believes it false, that there is no evidence for it and no imaginable reason it would be true.\nThe consensus of the survey articles (,, and ) is that the evidence for it is strong but not overwhelming, so that while it is probably true there is reasonable doubt.\nSome of the arguments for and against the Riemann hypothesis are listed by, , and , and include the following:\n\n## Popular expositions\nNote: Derbyshire 2003, Rockmore 2005, Sabbagh 2003a, Sabbagh 2003b, Sautoy 2003, and Watkins 2015 are non-technical.\nEdwards 1974, Patterson 1988, Borwein/Choi/Rooney/Weirathmueller 2008, Mazur/Stein 2015, Broughan 2017, and Nahin 2021 give mathematical introductions.\nTitchmarsh 1986, Ivić 1985, and Karatsuba/Voronin 1992 are advanced [monographs](https://bliptext.com/articles/monograph).","current_revision":"2fac4e58-7d63-4470-b962-029c24b91553","created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T11:10:51.636Z","updated_at":"2025-08-07T11:27:22.367Z","search_vector":"'hypothesi':2 'riemann':1","search_content":" \n \n\n# \nIn mathematics, the **Riemann hypothesis** is the [conjecture](https://bliptext.com/articles/conjecture) that the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) has its [zeros](https://bliptext.com/articles/root-of-a-function) only at the negative even integers and [complex numbers](https://bliptext.com/articles/complex-number) with [real part](https://bliptext.com/articles/real-part) 1⁄2.\nMany consider it to be the most important [unsolved problem](https://bliptext.com/articles/list-of-unsolved-problems-in-mathematics) in [pure mathematics](https://bliptext.com/articles/pure-mathematics).\nIt is of great interest in [number theory](https://bliptext.com/articles/number-theory) because it implies results about the distribution of [prime numbers](https://bliptext.com/articles/prime-numbers).\nIt was proposed by, after whom it is named.\nThe Riemann hypothesis and some of its generalizations, along with [Goldbach's conjecture](https://bliptext.com/articles/goldbach-s-conjecture) and the [twin prime conjecture](https://bliptext.com/articles/twin-prime-conjecture), make up [Hilbert's eighth problem](https://bliptext.com/articles/hilbert-s-eighth-problem) in [David Hilbert](https://bliptext.com/articles/david-hilbert)'s list of [twenty-three unsolved problems](https://bliptext.com/articles/hilbert-s-problems); it is also one of the [Millennium Prize Problems](https://bliptext.com/articles/millennium-prize-problems) of the [Clay Mathematics Institute](https://bliptext.com/articles/clay-mathematics-institute), which offers [US$](https://bliptext.com/articles/us)1 million for a solution to any of them.\nThe name is also used for some closely related analogues, such as the [Riemann hypothesis for curves over finite fields](https://bliptext.com/articles/riemann-hypothesis-for-curves-over-finite-fields).\nThe Riemann zeta [function](https://bliptext.com/articles/function-mathematics) ζ(s) is a [function](https://bliptext.com/articles/function-mathematics) whose [argument](https://bliptext.com/articles/argument-of-a-function) s may be any complex number other than 1, and whose values are also complex.\nIt has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros.\nThe zeta function is also zero for other values of s, which are called nontrivial zeros.\nThe Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:\nThus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers 1⁄2 + it, where t is a [real number](https://bliptext.com/articles/real-number) and i is the [imaginary unit](https://bliptext.com/articles/imaginary-unit).\n# [Riemann hypothesis | upload.wikimedia.org/wikipedia/commons/3/30/Riemann///zeta///function///absolute///value.png]\n\n# Riemann zeta function\nThe [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) is defined for complex s with real part greater than 1 by the [absolutely convergent](https://bliptext.com/articles/absolute-convergence) [infinite series](https://bliptext.com/articles/infinite-series)\n[Leonhard Euler](https://bliptext.com/articles/leonhard-euler) considered this series in the 1730s for real values of s, in conjunction with his solution to the [Basel problem](https://bliptext.com/articles/basel-problem).\nHe also proved that it equals the [Euler product](https://bliptext.com/articles/euler-product)\nwhere the [infinite product](https://bliptext.com/articles/infinite-product) extends over all prime numbers p.\nThe Riemann hypothesis discusses zeros outside the [region of convergence](https://bliptext.com/articles/region-of-convergence) of this series and Euler product.\nTo make sense of the hypothesis, it is necessary to [analytically continue](https://bliptext.com/articles/analytic-continuation) the function to obtain a form that is valid for all complex s.\nBecause the zeta function is [meromorphic](https://bliptext.com/articles/meromorphic), all choices of how to perform this analytic continuation will lead to the same result, by the [identity theorem](https://bliptext.com/articles/identity-theorem).\nA first step in this continuation observes that the series for the zeta function and the [Dirichlet eta function](https://bliptext.com/articles/dirichlet-eta-function) satisfy the relation\nwithin the region of convergence for both series.\nBut the eta function series on the right converges not just when the real part of s is greater than one, but more generally whenever s has positive real part.\nThus the zeta function can be redefined as, extending it from Re(s) > 1 to a larger domain: Re(s) > 0, except for the points where 1-2/2^s is zero.\nThese are the points where n can be any nonzero integer; the zeta function can be extended to these values too by taking limits (see ), giving a finite value for all values of s with positive real part except the [simple pole](https://bliptext.com/articles/simple-pole) at s = 1.\nIn the strip 0 1) by analytic continuation in the complex variable ρ in the region Re(ρ) > 0, i.e. they should be considered as Ei(ρ log x).\nThe other terms also correspond to zeros: the dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros.\nFor some graphs of the sums of the first few terms of this series see or.\nThis formula says that the zeros of the Riemann zeta function control the [oscillations](https://bliptext.com/articles/oscillation) of primes around their \"expected\" positions.\nRiemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1.\nHe checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.\n\"The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely, number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes.\"\n\n# Consequences\nThe practical uses of the Riemann hypothesis include many propositions known to be true under the Riemann hypothesis, and some that can be shown to be equivalent to the Riemann hypothesis.\n\n## Distribution of prime numbers\n[Riemann's explicit formula](https://bliptext.com/articles/riemann-s-explicit-formula) for [the number of primes less than a given number](https://bliptext.com/articles/prime-counting-function) states that, in terms of a sum over the zeros of the Riemann zeta function, the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function.\nIn particular, the error term in the [prime number theorem](https://bliptext.com/articles/prime-number-theorem) is closely related to the position of the zeros.\nFor example, if β is the [upper bound](https://bliptext.com/articles/upper-bound) of the real parts of the zeros, then\n, where \\pi(x) is the [prime-counting function](https://bliptext.com/articles/prime-counting-function), is the [logarithmic integral function](https://bliptext.com/articles/logarithmic-integral-function), \\log(x) is the [natural logarithm](https://bliptext.com/articles/natural-logarithm) of x, and [big O notation](https://bliptext.com/articles/big-o-notation) is used here.\nIt is already known that 1/2 ≤ β ≤ 1.\n[Von Koch (1901)](https://bliptext.com/articles/) proved that the Riemann hypothesis implies the \"best possible\" bound for the error of the prime number theorem.\nA precise version of von Koch's result, due to, says that the Riemann hypothesis implies\nalso showed that the Riemann hypothesis implies\nwhere \\psi(x) is [Chebyshev's second function](https://bliptext.com/articles/chebyshev-function).\nproved that the Riemann hypothesis implies that for all x \\geq 2 there is a prime p satisfying\nThe constant 4/π may\nbe reduced to (1 + ε) provided that x is taken to be sufficiently large.\nThis is an explicit version of a theorem of [Cramér](https://bliptext.com/articles/cram-r).\n\n## Growth of arithmetic functions\nThe Riemann hypothesis implies strong bounds on the growth of many other [arithmetic functions](https://bliptext.com/articles/arithmetic-function), in addition to the primes counting function above.\nOne example involves the [Möbius function](https://bliptext.com/articles/m-bius-function) μ.\nThe statement that the equation\nis valid for every s with real part greater than 1/2, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis.\nFrom this we can also conclude that if the [Mertens function](https://bliptext.com/articles/mertens-function) is defined by\nthen the claim that\nfor every positive ε is equivalent to the Riemann hypothesis ([J. E. Littlewood](https://bliptext.com/articles/john-edensor-littlewood), 1912; see for instance: paragraph 14.25 in ).\nThe [determinant](https://bliptext.com/articles/determinant) of the order n [Redheffer matrix](https://bliptext.com/articles/redheffer-matrix) is equal to M(n), so the Riemann hypothesis can also be stated as a condition on the growth of these [determinant](https://bliptext.com/articles/determinant)s.\nLittlewood's result has been improved several times since then, by [Edmund Landau](https://bliptext.com/articles/edmund-landau), [Edward Charles Titchmarsh](https://bliptext.com/articles/edward-charles-titchmarsh), Helmut Maier and [Hugh Montgomery](https://bliptext.com/articles/hugh-lowell-montgomery), and [Kannan Soundararajan](https://bliptext.com/articles/kannan-soundararajan).\nSoundararajan's result is that, conditional on the Riemann hypothesis,\nThe Riemann hypothesis puts a rather tight bound on the growth of M, since disproved the slightly stronger [Mertens conjecture](https://bliptext.com/articles/mertens-conjecture)\nAnother closely related result is due to, that the Riemann hypothesis is equivalent to the statement that the [Euler characteristic](https://bliptext.com/articles/euler-characteristic) of the [simplicial complex](https://bliptext.com/articles/simplicial-complex) determined by the lattice of integers under divisibility is for all \\epsilon>0 (see [incidence algebra](https://bliptext.com/articles/incidence-algebra)).\nThe Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic functions aside from μ(n).\nA typical example is [Robin's theorem](https://bliptext.com/articles/robin-s-theorem), which states that if σ(n) is the [sigma function](https://bliptext.com/articles/divisor-function), given by\nthen\nfor all n > 5040 if and only if the Riemann hypothesis is true, where γ is the [Euler–Mascheroni constant](https://bliptext.com/articles/euler-mascheroni-constant).\nA related bound was given by [Jeffrey Lagarias](https://bliptext.com/articles/jeffrey-lagarias) in 2002, who proved that the Riemann hypothesis is equivalent to the statement that:\nfor every [natural number](https://bliptext.com/articles/natural-number) n > 1, where H_n is the nth [harmonic number](https://bliptext.com/articles/harmonic-number).\nThe Riemann hypothesis is also true if and only if the inequality\nis true for all n ≥ 120569# where φ(n) is [Euler's totient function](https://bliptext.com/articles/euler-s-totient-function) and 120569# is the [product of the first](https://bliptext.com/articles/primorial) 120569 primes.\nAnother example was found by [Jérôme Franel](https://bliptext.com/articles/j-r-me-franel), and extended by [Landau](https://bliptext.com/articles/edmund-landau) (see ).\nThe Riemann hypothesis is equivalent to several statements showing that the terms of the [Farey sequence](https://bliptext.com/articles/farey-sequence) are fairly regular.\nOne such equivalence is as follows: if Fn is the Farey sequence of order n, beginning with 1/n and up to 1/1, then the claim that for all ε > 0\nis equivalent to the Riemann hypothesis.\nHere\nis the number of terms in the Farey sequence of order n.\nFor an example from [group theory](https://bliptext.com/articles/group-theory), if g(n) is [Landau's function](https://bliptext.com/articles/landau-s-function) given by the maximal order of elements of the [symmetric group](https://bliptext.com/articles/symmetric-group) Sn of degree n, then showed that the Riemann hypothesis is equivalent to the bound\nfor all sufficiently large n.\n\n## Lindelöf hypothesis and growth of the zeta function\nThe Riemann hypothesis has various weaker consequences as well; one is the [Lindelöf hypothesis](https://bliptext.com/articles/lindel-f-hypothesis) on the rate of growth of the zeta function on the critical line, which says that, for any ε > 0,\nas.\nThe Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip.\nFor example, it implies that\nso the growth rate of ζ(1 + it) and its inverse would be known up to a factor of 2.\n\n## Large prime gap conjecture\nThe prime number theorem implies that on average, the [gap](https://bliptext.com/articles/prime-gap) between the prime p and its successor is log p.\nHowever, some gaps between primes may be much larger than the average.\n[Cramér](https://bliptext.com/articles/harald-cram-r) proved that, assuming the Riemann hypothesis, every gap is O(√p log p).\nThis is a case in which even the best bound that can be proved using the Riemann hypothesis is far weaker than what seems true: [Cramér's conjecture](https://bliptext.com/articles/cram-r-s-conjecture) implies that every gap is O((log p)2), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis.\nNumerical evidence supports Cramér's conjecture.\n\n## Analytic criteria equivalent to the Riemann hypothesis\nMany statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in proving (or disproving) it.\nSome typical examples are as follows.\n(Others involve the [divisor function](https://bliptext.com/articles/divisor-function) σ(n).)\nThe [Riesz criterion](https://bliptext.com/articles/riesz-criterion) was given by, to the effect that the bound\nholds for all ε > 0 if and only if the Riemann hypothesis holds.\nSee also the [Hardy–Littlewood criterion](https://bliptext.com/articles/riesz-function).\nproved that the Riemann hypothesis is true if and only if the space of functions of the form\nwhere ρ(z) is the fractional part of z, 0 ≤ θν ≤ 1, and\nis dense in the Hilbert space L2(0,1) of square-integrable functions on the unit interval.\nextended this by showing that the zeta function has no zeros with real part greater than 1/p if and only if this function space is dense in Lp(0,1).\nThis Nyman-Beurling criterion was strengthened by Baez-Duarte to the case where.\nshowed that the Riemann hypothesis is true if and only if the integral equation\nhas no non-trivial bounded solutions \\varphi for.\n[Weil's criterion](https://bliptext.com/articles/weil-s-criterion) is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis.\nRelated is [Li's criterion](https://bliptext.com/articles/li-s-criterion), a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis.\nproved that the Riemann hypothesis is equivalent to the statement that \\zeta'(s), the derivative of \\zeta(s), has no zeros in the strip\nThat \\zeta(s) has only simple zeros on the critical line is equivalent to its derivative having no zeros on the critical line.\nThe [Farey sequence](https://bliptext.com/articles/farey-sequence) provides two equivalences, due to [Jerome Franel](https://bliptext.com/articles/jerome-franel) and [Edmund Landau](https://bliptext.com/articles/edmund-landau) in 1924.\nThe [de Bruijn–Newman constant](https://bliptext.com/articles/de-bruijn-newman-constant) denoted by **Λ** and named after [Nicolaas Govert de Bruijn](https://bliptext.com/articles/nicolaas-govert-de-bruijn) and [Charles M. Newman](https://bliptext.com/articles/charles-m-newman), is defined\nas the unique real number such that the [function](https://bliptext.com/articles/function-mathematics)\nthat is parametrised by a [real](https://bliptext.com/articles/real-number) parameter λ, has a [complex](https://bliptext.com/articles/complex-number) variable z and is defined using a super-exponentially decaying function\nhas only real zeros if and only if λ ≥ Λ.\nSince the Riemann hypothesis is equivalent to the claim that all the zeroes of H(0, z) are real, the Riemann hypothesis is equivalent to the conjecture that.\nBrad Rodgers and [Terence Tao](https://bliptext.com/articles/terence-tao) discovered the equivalence is actually \\Lambda = 0 by proving zero to be the lower bound of the constant.\nProving zero is also the upper bound would therefore prove the Riemann hypothesis.\nAs of April 2020 the upper bound is.\n\n## Consequences of the generalized Riemann hypothesis\nSeveral applications use the [generalized Riemann hypothesis](https://bliptext.com/articles/generalized-riemann-hypothesis) for [Dirichlet L-series](https://bliptext.com/articles/dirichlet-l-series) or [zeta functions of number fields](https://bliptext.com/articles/dedekind-zeta-function) rather than just the Riemann hypothesis.\nMany basic properties of the Riemann zeta function can easily be generalized to all Dirichlet L-series, so it is plausible that a method that proves the Riemann hypothesis for the Riemann zeta function would also work for the generalized Riemann hypothesis for Dirichlet L-functions.\nSeveral results first proved using the generalized Riemann hypothesis were later given unconditional proofs without using it, though these were usually much harder.\nMany of the consequences on the following list are taken from.\n\n## Excluded middle\nSome consequences of the RH are also consequences of its negation, and are thus theorems.\nIn their discussion of the [Hecke, Deuring, Mordell, Heilbronn theorem](https://bliptext.com/articles/), say\n\"The method of proof here is truly amazing. If the generalized Riemann hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!!\"\nCare should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample.\n\n### Littlewood's theorem\nThis concerns the sign of the error in the [prime number theorem](https://bliptext.com/articles/prime-number-theorem).\nIt has been computed that π(x) li(x).\nIn 1914 Littlewood proved that there are arbitrarily large values of x for which\nand that there are also arbitrarily large values of x for which\nThus the difference π(x) − li(x) changes sign infinitely many times.\n[Skewes' number](https://bliptext.com/articles/skewes-number) is an estimate of the value of x corresponding to the first sign change.\nLittlewood's proof is divided into two cases: the RH is assumed false (about half a page of ), and the RH is assumed true (about a dozen pages).\nfollowed this up with a paper on the number of times \\Delta(n) changes sign in the interval \\Delta(n).\n\n### Gauss's class number conjecture\nThis is the [conjecture](https://bliptext.com/articles/class-number-problem) (first stated in article 303 of Gauss's [Disquisitiones Arithmeticae](https://bliptext.com/articles/disquisitiones-arithmeticae)) that there are only finitely many imaginary quadratic fields with a given class number.\nOne way to prove it would be to show that as the discriminant\nD → −∞\nthe class number\nh(D) → ∞\n.\nThe following sequence of theorems involving the Riemann hypothesis is described in :\n(In the work of Hecke and Heilbronn, the only L-functions that occur are those attached to imaginary quadratic characters, and it is only for those L-functions that GRH is true or GRH is false is intended; a failure of GRH for the L-function of a cubic Dirichlet character would, strictly speaking, mean GRH is false, but that was not the kind of failure of GRH that Heilbronn had in mind, so his assumption was more restricted than simply GRH is false.)\nIn 1935, [Carl Siegel](https://bliptext.com/articles/carl-siegel) strengthened the result without using RH or GRH in any way.\n\n### Growth of Euler's totient\nIn 1983 [J. L. Nicolas](https://bliptext.com/articles/jean-louis-nicolas) proved that\nfor infinitely many n, where φ(n) is [Euler's totient function](https://bliptext.com/articles/euler-s-totient-function) and γ is [Euler's constant](https://bliptext.com/articles/euler-s-constant).\nRibenboim remarks that: \"The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption.\"\n\n# Generalizations and analogs\n\n## Dirichlet L-series and other number fields\nThe Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, but much more general, global [L-functions](https://bliptext.com/articles/l-function).\nIn this broader setting, one expects the non-trivial zeros of the global L-functions to have real part 1/2.\nIt is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta function, which account for the true importance of the Riemann hypothesis in mathematics.\nThe [generalized Riemann hypothesis](https://bliptext.com/articles/generalized-riemann-hypothesis) extends the Riemann hypothesis to all [Dirichlet L-functions](https://bliptext.com/articles/dirichlet-l-function).\nIn particular it implies the conjecture that [Siegel zeros](https://bliptext.com/articles/siegel-zero) (zeros of L-functions between 1/2 and 1) do not exist.\nThe [extended Riemann hypothesis](https://bliptext.com/articles/extended-riemann-hypothesis) extends the Riemann hypothesis to all [Dedekind zeta functions](https://bliptext.com/articles/dedekind-zeta-function) of [algebraic number fields](https://bliptext.com/articles/algebraic-number-field).\nThe extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis.\nThe Riemann hypothesis can also be extended to the L-functions of [Hecke characters](https://bliptext.com/articles/hecke-character) of number fields.\nThe [grand Riemann hypothesis](https://bliptext.com/articles/grand-riemann-hypothesis) extends it to all [automorphic zeta functions](https://bliptext.com/articles/automorphic-l-function), such as [Mellin transforms](https://bliptext.com/articles/mellin-transform) of [Hecke eigenforms](https://bliptext.com/articles/hecke-eigenform).\n\n## Function fields and zeta functions of varieties over finite fields\nintroduced global zeta functions of (quadratic) [function fields](https://bliptext.com/articles/function-field-of-an-algebraic-variety) and conjectured an analogue of the Riemann hypothesis for them, which has been proved by Hasse in the genus 1 case and by in general.\nFor instance, the fact that the [Gauss sum](https://bliptext.com/articles/quadratic-gauss-sum), of the quadratic character of a [finite field](https://bliptext.com/articles/finite-field) of size q (with q odd), has absolute value \\sqrt{q} is actually an instance of the Riemann hypothesis in the function field setting.\nThis led to conjecture a similar statement for all [algebraic varieties](https://bliptext.com/articles/algebraic-variety); the resulting [Weil conjectures](https://bliptext.com/articles/weil-conjectures) were proved by.\n\n## Arithmetic zeta functions of arithmetic schemes and their L-factors\n[Arithmetic zeta functions](https://bliptext.com/articles/arithmetic-zeta-function) generalise the Riemann and Dedekind zeta functions as well as the zeta functions of varieties over finite fields to every arithmetic scheme or a scheme of finite type over integers.\nThe arithmetic zeta function of a regular connected [equidimensional](https://bliptext.com/articles/equidimensionality) arithmetic scheme of Kronecker dimension n can be factorized into the product of appropriately defined L-factors and an auxiliary factor.\nAssuming a functional equation and meromorphic continuation, the generalized Riemann hypothesis for the L-factor states that its zeros inside the critical strip lie on the central line.\nCorrespondingly, the generalized Riemann hypothesis for the arithmetic zeta function of a regular connected equidimensional arithmetic scheme states that its zeros inside the critical strip lie on vertical lines and its poles inside the critical strip lie on vertical lines.\nThis is known for schemes in positive characteristic and follows from, but remains entirely unknown in characteristic zero.\n\n## Selberg zeta functions\nintroduced the [Selberg zeta function](https://bliptext.com/articles/selberg-zeta-function) of a Riemann surface.\nThese are similar to the Riemann zeta function: they have a functional equation, and an infinite product similar to the Euler product but taken over closed geodesics rather than primes.\nThe [Selberg trace formula](https://bliptext.com/articles/selberg-trace-formula) is the analogue for these functions of the [explicit formulas](https://bliptext.com/articles/explicit-formula-l-function) in prime number theory.\nSelberg proved that the Selberg zeta functions satisfy the analogue of the Riemann hypothesis, with the imaginary parts of their zeros related to the [eigenvalues](https://bliptext.com/articles/eigenvalue) of the Laplacian operator of the Riemann surface.\n\n## Ihara zeta functions\nThe [Ihara zeta function](https://bliptext.com/articles/ihara-zeta-function) of a finite graph is an analogue of the [Selberg zeta function](https://bliptext.com/articles/selberg-zeta-function), which was first introduced by [Yasutaka Ihara](https://bliptext.com/articles/yasutaka-ihara) in the context of discrete subgroups of the two-by-two p-adic special linear group.\nA regular finite graph is a [Ramanujan graph](https://bliptext.com/articles/ramanujan-graph), a mathematical model of efficient communication networks, if and only if its Ihara zeta function satisfies the analogue of the Riemann hypothesis as was pointed out by [T. Sunada](https://bliptext.com/articles/toshikazu-sunada).\n\n## Montgomery's pair correlation conjecture\nsuggested the [pair correlation conjecture](https://bliptext.com/articles/pair-correlation-conjecture) that the correlation functions of the (suitably normalized) zeros of the zeta function should be the same as those of the eigenvalues of a [random hermitian matrix](https://bliptext.com/articles/gaussian-unitary-ensemble).\nshowed that this is supported by large-scale numerical calculations of these correlation functions.\nMontgomery showed that (assuming the Riemann hypothesis) at least 2/3 of all zeros are simple, and a related conjecture is that all zeros of the zeta function are simple (or more generally have no non-trivial integer linear relations between their imaginary parts).\n[Dedekind zeta functions](https://bliptext.com/articles/dedekind-zeta-function) of algebraic number fields, which generalize the Riemann zeta function, often do have multiple complex zeros.\nThis is because the Dedekind zeta functions factorize as a product of powers of [Artin L-functions](https://bliptext.com/articles/artin-l-function), so zeros of [Artin L-functions](https://bliptext.com/articles/artin-l-function) sometimes give rise to multiple zeros of Dedekind zeta functions.\nOther examples of zeta functions with multiple zeros are the L-functions of some [elliptic curves](https://bliptext.com/articles/elliptic-curve): these can have multiple zeros at the real point of their critical line; the [Birch-Swinnerton-Dyer conjecture](https://bliptext.com/articles/birch-swinnerton-dyer-conjecture) predicts that the multiplicity of this zero is the rank of the elliptic curve.\n\n## Other zeta functions\nThere are [many other examples](https://bliptext.com/articles/zeta-function) of zeta functions with analogues of the Riemann hypothesis, some of which have been proved.\n[Goss zeta functions](https://bliptext.com/articles/goss-zeta-function) of function fields have a Riemann hypothesis, proved by.\n[The main conjecture](https://bliptext.com/articles/main-conjecture-of-iwasawa-theory) of [Iwasawa theory](https://bliptext.com/articles/iwasawa-theory), proved by [Barry Mazur](https://bliptext.com/articles/barry-mazur) and [Andrew Wiles](https://bliptext.com/articles/andrew-wiles) for [cyclotomic fields](https://bliptext.com/articles/cyclotomic-field), and Wiles for [totally real fields](https://bliptext.com/articles/totally-real-number-field), identifies the zeros of a p-adic L-function with the eigenvalues of an operator, so can be thought of as an analogue of the [Hilbert–Pólya conjecture](https://bliptext.com/articles/hilbert-p-lya-conjecture) for p-adic L-functions.\n\n# Attempted proofs\nSeveral mathematicians have addressed the Riemann hypothesis, but none of their attempts has yet been accepted as a proof.\nlists some incorrect solutions.\n\n## Operator theory\nHilbert and Pólya suggested that one way to derive the Riemann hypothesis would be to find a [self-adjoint operator](https://bliptext.com/articles/self-adjoint-operator), from the existence of which the statement on the real parts of the zeros of ζ(s) would follow when one applies the criterion on real [eigenvalues](https://bliptext.com/articles/eigenvalue).\nSome support for this idea comes from several analogues of the Riemann zeta functions whose zeros correspond to eigenvalues of some operator: the zeros of a zeta function of a variety over a finite field correspond to eigenvalues of a [Frobenius element](https://bliptext.com/articles/frobenius-element) on an [étale cohomology](https://bliptext.com/articles/tale-cohomology) group, the zeros of a [Selberg zeta function](https://bliptext.com/articles/selberg-zeta-function) are eigenvalues of a [Laplacian operator](https://bliptext.com/articles/laplacian-operator) of a Riemann surface, and the zeros of a [p-adic zeta function](https://bliptext.com/articles/p-adic-zeta-function) correspond to eigenvectors of a Galois action on [ideal class groups](https://bliptext.com/articles/ideal-class-group).\nshowed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of [random matrices](https://bliptext.com/articles/random-matrices) drawn from the [Gaussian unitary ensemble](https://bliptext.com/articles/gaussian-unitary-ensemble).\nThis gives some support to the [Hilbert–Pólya conjecture](https://bliptext.com/articles/hilbert-p-lya-conjecture).\nIn 1999, [Michael Berry](https://bliptext.com/articles/michael-berry-physicist) and [Jonathan Keating](https://bliptext.com/articles/jonathan-keating) conjectured that there is some unknown quantization \\hat H of the classical Hamiltonian H = xp so that\nand even more strongly, that the Riemann zeros coincide with the spectrum of the operator.\nThis is in contrast to [canonical quantization](https://bliptext.com/articles/canonical-quantization), which leads to the [Heisenberg uncertainty principle](https://bliptext.com/articles/heisenberg-uncertainty-principle) and the [natural numbers](https://bliptext.com/articles/natural-numbers) as spectrum of the [quantum harmonic oscillator](https://bliptext.com/articles/quantum-harmonic-oscillator).\nThe crucial point is that the Hamiltonian should be a self-adjoint operator so that the quantization would be a realization of the Hilbert–Pólya program.\nIn a connection with this quantum mechanical problem Berry and Connes had proposed that the inverse of the potential of the Hamiltonian is connected to the [half-derivative](https://bliptext.com/articles/half-derivative) of the function\nthen, in Hilbert-Polya approach\nThis yields a Hamiltonian whose eigenvalues are the square of the imaginary part of the Riemann zeros, and also that the [functional determinant](https://bliptext.com/articles/functional-determinant) of this [Hamiltonian operator](https://bliptext.com/articles/hamiltonian-operator) is just the [Riemann Xi function](https://bliptext.com/articles/riemann-xi-function).\nIn fact the Riemann Xi function would be proportional to the functional determinant ([Hadamard product](https://bliptext.com/articles/hadamard-product-matrices))\nHowever this operator is not useful in practice since it includes the inverse function (implicit function ) of the potential but not the potential itself.\nThe analogy with the Riemann hypothesis over [finite fields](https://bliptext.com/articles/finite-field) suggests that the Hilbert space containing eigenvectors corresponding to the zeros might be some sort of first [cohomology group](https://bliptext.com/articles/cohomology-group) of the [spectrum](https://bliptext.com/articles/spectrum-of-a-ring) Spec (Z) of the integers.\ndescribed some of the attempts to find such a cohomology theory.\nconstructed a natural space of invariant functions on the upper half plane that has eigenvalues under the Laplacian operator that correspond to zeros of the Riemann zeta function—and remarked that in the unlikely event that one could show the existence of a suitable positive definite inner product on this space, the Riemann hypothesis would follow.\ndiscussed a related example, where due to a bizarre bug a computer program listed zeros of the Riemann zeta function as eigenvalues of the same [Laplacian operator](https://bliptext.com/articles/laplacian-operator).\nsurveyed some of the attempts to construct a suitable physical model related to the Riemann zeta function.\n\n## Lee–Yang theorem\nThe [Lee–Yang theorem](https://bliptext.com/articles/lee-yang-theorem) states that the zeros of certain [partition functions](https://bliptext.com/articles/partition-function-statistical-mechanics) in [statistical mechanics](https://bliptext.com/articles/statistical-mechanics) all lie on a \"critical line\" with their real part equal to 0, and this has led to some speculation about a relationship with the Riemann hypothesis.\n\n## Turán's result\nshowed that if the functions\nhave no zeros when the real part of s is greater than one then\nwhere λ(n) is the [Liouville function](https://bliptext.com/articles/liouville-function) given by (−1)r if n has r prime factors.\nHe showed that this in turn would imply that the Riemann hypothesis is true.\nBut proved that T(x) is negative for infinitely many x (and also disproved the closely related [Pólya conjecture](https://bliptext.com/articles/p-lya-conjecture)), and showed that the smallest such x is 72 185 376 951 205.\nshowed by numerical calculation that the finite [Dirichlet series](https://bliptext.com/articles/dirichlet-series) above for N=19 has a zero with real part greater than 1.\nTurán also showed that a somewhat weaker assumption, the nonexistence of zeros with real part greater than 1+N−1/2+ε for large N in the finite Dirichlet series above, would also imply the Riemann hypothesis, but showed that for all sufficiently large N these series have zeros with real part greater than 1 + (log log N)/(4 log N).\nTherefore, Turán's result is [vacuously true](https://bliptext.com/articles/vacuous-truth) and cannot help prove the Riemann hypothesis.\n\n## Noncommutative geometry\nhas described a relationship between the Riemann hypothesis and [noncommutative geometry](https://bliptext.com/articles/noncommutative-geometry), and showed that a suitable analog of the [Selberg trace formula](https://bliptext.com/articles/selberg-trace-formula) for the action of the [idèle class group](https://bliptext.com/articles/id-le-class-group) on the adèle class space would imply the Riemann hypothesis.\nSome of these ideas are elaborated in.\n\n## Hilbert spaces of entire functions\nshowed that the Riemann hypothesis would follow from a positivity condition on a certain [Hilbert space](https://bliptext.com/articles/hilbert-space) of [entire functions](https://bliptext.com/articles/entire-function).\nHowever showed that the necessary positivity conditions are not satisfied.\nDespite this obstacle, de Branges has continued to work on an attempted proof of the Riemann hypothesis along the same lines, but this has not been widely accepted by other mathematicians.\n\n## Quasicrystals\nThe Riemann hypothesis implies that the zeros of the zeta function form a [quasicrystal](https://bliptext.com/articles/quasicrystal), a distribution with discrete support whose [Fourier transform](https://bliptext.com/articles/fourier-transform) also has discrete support.\nsuggested trying to prove the Riemann hypothesis by classifying, or at least studying, 1-dimensional quasicrystals.\n\n## Arithmetic zeta functions of models of elliptic curves over number fields\nWhen one goes from geometric dimension one, e.g. an [algebraic number field](https://bliptext.com/articles/algebraic-number-field), to geometric dimension two, e.g. a regular model of an [elliptic curve](https://bliptext.com/articles/elliptic-curve) over a number field, the two-dimensional part of the generalized Riemann hypothesis for the [arithmetic zeta function](https://bliptext.com/articles/arithmetic-zeta-function) of the model deals with the poles of the zeta function.\nIn dimension one the study of the zeta integral in [Tate's thesis](https://bliptext.com/articles/tate-s-thesis) does not lead to new important information on the Riemann hypothesis.\nContrary to this, in dimension two work of [Ivan Fesenko](https://bliptext.com/articles/ivan-fesenko) on two-dimensional generalisation of Tate's thesis includes an integral representation of a zeta integral closely related to the zeta function.\nIn this new situation, not possible in dimension one, the poles of the zeta function can be studied via the zeta integral and associated adele groups.\nRelated conjecture of on the positivity of the fourth derivative of a boundary function associated to the zeta integral essentially implies the pole part of the generalized Riemann hypothesis.\nproved that the latter, together with some technical assumptions, implies Fesenko's conjecture.\n\n## Multiple zeta functions\nDeligne's proof of the Riemann hypothesis over finite fields used the zeta functions of product varieties, whose zeros and poles correspond to sums of zeros and poles of the original zeta function, in order to bound the real parts of the zeros of the original zeta function.\nBy analogy, introduced multiple zeta functions whose zeros and poles correspond to sums of zeros and poles of the Riemann zeta function.\nTo make the series converge he restricted to sums of zeros or poles all with non-negative imaginary part.\nSo far, the known bounds on the zeros and poles of the multiple zeta functions are not strong enough to give useful estimates for the zeros of the Riemann zeta function.\n\n# Location of the zeros\n\n## Number of zeros\nThe functional equation combined with the [argument principle](https://bliptext.com/articles/argument-principle) implies that the number of zeros of the zeta function with imaginary part between 0 and T is given by for s=1/2+iT, where the argument is defined by varying it continuously along the line with Im(s)=T, starting with argument 0 at ∞+iT.\nThis is the sum of a large but well understood term\nand a small but rather mysterious term\nSo the density of zeros with imaginary part near T is about log(T)/(2π), and the function S describes the small deviations from this.\nThe function S(t) jumps by 1 at each zero of the zeta function, and for t ≥ 8 it decreases [monotonically](https://bliptext.com/articles/monotonically-decreasing) between zeros with derivative close to −log t.\nproved that, if T > e, then\n[Karatsuba](https://bliptext.com/articles/anatolii-alexeevitch-karatsuba) (1996) proved that every interval (T, T+H] for contains at least points where the function S(t) changes sign.\nshowed that the average moments of even powers of S are given by\nThis suggests that S(T)/(log log T)1/2 resembles a [Gaussian random variable](https://bliptext.com/articles/gaussian-random-variable) with mean 0 and variance 2π2 ( proved this fact).\nIn particular |S(T)| is usually somewhere around (log log T)1/2, but occasionally much larger.\nThe exact order of growth of S(T) is not known.\nThere has been no unconditional improvement to Riemann's original bound S(T)=O(log T), though the Riemann hypothesis implies the slightly smaller bound S(T)=O(log T/log log T).\nThe true order of magnitude may be somewhat less than this, as random functions with the same distribution as S(T) tend to have growth of order about log(T)1/2.\nIn the other direction it cannot be too small: showed that S(T) ≠ o((log T)1/3/(log log T)7/3), and assuming the Riemann hypothesis Montgomery showed that S(T) ≠ o((log T)1/2/(log log T)1/2).\nNumerical calculations confirm that S grows very slowly: |S(T)| < 1 for T 1, t real, and looking at the limit as σ → 1.\nThis inequality follows by taking the real part of the log of the Euler product to see that\nwhere the sum is over all prime powers pn, so that\nwhich is at least 1 because all the terms in the sum are positive, due to the inequality\n\n## Zero-free regions\nThe most extensive computer search by Platt and Trudgian for counter examples of the Riemann hypothesis has verified it for.\nBeyond that zero-free regions are known as inequalities concerning σ + it, which can be zeroes.\nThe oldest version is from [De la Vallée-Poussin (1899–1900)](https://bliptext.com/articles/), who proved there is a region without zeroes that satisfies 1 − σ ≥ C⁄log(t) for some positive constant C.\nIn other words, zeros cannot be too close to the line there is a zero-free region close to this line.\nThis has been enlarged by several authors using methods such as [Vinogradov's mean-value theorem](https://bliptext.com/articles/vinogradov-s-mean-value-theorem).\nThe most recent paper by Mossinghoff, Trudgian and Yang is from December 2022 and provides four zero-free regions that improved the previous results of Kevin Ford from 2002, Mossinghoff and Trudgian themselves from 2015 and Pace Nielsen's slight improvement of Ford from October 2022:\nThe paper also presents an improvement to the second zero-free region, whose bounds are unknown on account of |t| being merely assumed to be \"sufficiently large\" to fulfill the requirements of the paper's proof.\nThis region is\n.\n\n# Zeros on the critical line\nand showed there are infinitely many zeros on the critical line, by considering moments of certain functions related to the zeta function.\nproved that at least a (small) positive proportion of zeros lie on the line.\nimproved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, and improved this further to two-fifths.\nIn 2020, this estimate was extended to five-twelfths by Pratt, Robles, [Zaharescu](https://bliptext.com/articles/alexandru-zaharescu) and Zeindler by considering extended mollifiers that can accommodate higher order derivatives of the zeta function and their associated Kloosterman sums.\nMost zeros lie close to the critical line.\nMore precisely, showed that for any positive ε, the number of zeroes with real part at least 1/2+ε and imaginary part at between -T and T is O(T).\nCombined with the facts that zeroes on the critical strip are symmetric about the critical line and that the total number of zeroes in the critical strip is, [almost all](https://bliptext.com/articles/almost-all) non-trivial zeroes are within a distance ε of the critical line.\ngives several more precise versions of this result, called zero density estimates, which bound the number of zeros in regions with imaginary part at most T and real part at least 1/2+ε.\n\n## Hardy–Littlewood conjectures\nIn 1914 [Godfrey Harold Hardy](https://bliptext.com/articles/g-h-hardy) proved that has infinitely many real zeros.\nThe next two conjectures of [Hardy](https://bliptext.com/articles/g-h-hardy) and [John Edensor Littlewood](https://bliptext.com/articles/john-edensor-littlewood) on the distance between real zeros of and on the density of zeros of on the interval (T,T+H] for sufficiently large T > 0, and and with as small as possible value of a > 0, where is an arbitrarily small number, open two new directions in the investigation of the Riemann zeta function:\nLet N(T) be the total number of real zeros, and N_0(T) be the total number of zeros of odd order of the function lying on the interval (0,T]~.\n\n## Selberg's zeta function conjecture\ninvestigated the problem of Hardy–Littlewood 2 and proved that for any ε > 0 there exists such and c = c(ε) > 0, such that for T \\geq T_0 and the inequality is true.\nSelberg conjectured that this could be tightened to H=T^{0.5}.\nproved that for a fixed ε satisfying the condition 0 < ε < 0.001, a sufficiently large T and, , the interval (T, T+H) contains at least cH log(T) real zeros of the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) and therefore confirmed the Selberg conjecture.\nThe estimates of Selberg and Karatsuba can not be improved in respect of the order of growth as T → ∞.\nproved that an analog of the Selberg conjecture holds for almost all intervals (T, T+H],, where ε is an arbitrarily small fixed positive number. The Karatsuba method permits to investigate zeros of the Riemann zeta function on \"supershort\" intervals of the critical line, that is, on the intervals (T, T+H], the length H of which grows slower than any, even arbitrarily small degree T.\nIn particular, he proved that for any given numbers ε, satisfying the conditions almost all intervals (T, T+H] for contain at least zeros of the function . This estimate is quite close to the one that follows from the Riemann hypothesis.\n\n## Numerical calculations\nThe function\nhas the same zeros as the zeta function in the critical strip, and is real on the critical line because of the functional equation, so one can prove the existence of zeros exactly on the real line between two points by checking numerically that the function has opposite signs at these points.\nUsually one writes\nwhere Hardy's [Z function](https://bliptext.com/articles/z-function) and the [Riemann–Siegel theta function](https://bliptext.com/articles/riemann-siegel-theta-function) θ are uniquely defined by this and the condition that they are smooth real functions with θ(0) = 0.\nBy finding many intervals where the function Z changes sign one can show that there are many zeros on the critical line.\nTo verify the Riemann hypothesis up to a given [imaginary part](https://bliptext.com/articles/imaginary-part) T of the zeros, one also has to check that there are no further zeros off the line in this region.\nThis can be done by calculating the total number of zeros in the region using [Turing's method](https://bliptext.com/articles/turing-s-method) and checking that it is the same as the number of zeros found on the line.\nThis allows one to verify the Riemann hypothesis computationally up to any desired value of T (provided all the zeros of the zeta function in this region are simple and on the critical line).\nThese calculations can also be used to estimate \\pi(x) for finite ranges of x. For example, using the latest result from 2020 (zeros up to height ), it has been shown that\nIn general, this inequality holds if where T is the largest known value such that the Riemann hypothesis is true for all zeros \\rho with.\nSome calculations of zeros of the zeta function are listed below, where the \"height\" of a zero is the magnitude of its imaginary part, and the height of the nth zero is denoted by γn. So far all zeros that have been checked are on the critical line and are simple.\n(A multiple zero would cause problems for the zero finding algorithms, which depend on finding sign changes between zeros.) For tables of the zeros, see or.\n\n## Gram points\nA [Gram point](https://bliptext.com/articles/gram-point) is a point on the critical line 1/2 + it where the zeta function is real and non-zero.\nUsing the expression for the zeta function on the critical line, ζ(1/2 + it) = Z(t)e− iθ(t), where Hardy's function, Z, is real for real t, and θ is the [Riemann–Siegel theta function](https://bliptext.com/articles/riemann-siegel-theta-function), we see that zeta is real when sin(θ(t)) = 0.\nThis implies that θ(t) is an integer multiple of π, which allows for the location of Gram points to be calculated fairly easily by inverting the formula for θ.\nThey are usually numbered as gn for n = 0, 1, ..., where gn is the unique solution of θ(t) = nπ.\nGram observed that there was often exactly one zero of the zeta function between any two Gram points; Hutchinson called this observation **[Gram's law](https://bliptext.com/articles/gram-s-law)**.\nThere are several other closely related statements that are also sometimes called Gram's law: for example, (−1)nZ(gn) is usually positive, or Z(t) usually has opposite sign at consecutive Gram points.\nThe imaginary parts γn of the first few zeros (in blue) and the first few Gram points gn are given in the following table\nThe first failure of Gram's law occurs at the 127th zero and the Gram point g126, which are in the \"wrong\" order.\nA Gram point t is called good if the zeta function is positive at 1/2 + it.\nThe indices of the \"bad\" Gram points where Z has the \"wrong\" sign are 126, 134, 195, 211, ... . A Gram block is an interval bounded by two good Gram points such that all the Gram points between them are bad.\nA refinement of Gram's law called Rosser's rule due to says that Gram blocks often have the expected number of zeros in them (the same as the number of Gram intervals), even though some of the individual Gram intervals in the block may not have exactly one zero in them.\nFor example, the interval bounded by g125 and g127 is a Gram block containing a unique bad Gram point g126, and contains the expected number 2 of zeros although neither of its two Gram intervals contains a unique zero.\nRosser et al. checked that there were no exceptions to Rosser's rule in the first 3 million zeros, although there are infinitely many exceptions to Rosser's rule over the entire zeta function.\nGram's rule and Rosser's rule both say that in some sense zeros do not stray too far from their expected positions.\nThe distance of a zero from its expected position is controlled by the function S defined above, which grows extremely slowly: its average value is of the order of (log log T)1/2, which only reaches 2 for T around 1024.\nThis means that both rules hold most of the time for small T but eventually break down often.\nIndeed, showed that both Gram's law and Rosser's rule fail in a positive proportion of cases.\nTo be specific, it is expected that in about 66% one zero is enclosed by two successive Gram points, but in 17% no zero and in 17% two zeros are in such a Gram-interval on the long run.\n# [This is a polar plot of the first 20 real values rn of the zeta function along the critical line, ζ(1/2 + it), with t running from 0 to 50. The values of rn in this range are the first 10 non-trivial Riemann zeta function zeros and the first 10 Gram points, each labeled by n. Fifty red points have been plotted between each rn, and the zeros are projected onto concentric magenta rings scaled to show the relative distance between their values of t.\nGram's law states that the curve usually crosses the real axis once between zeros. | upload.wikimedia.org/wikipedia/commons/f/fc/RiemannZeta///Zeros.svg]\n\n# Arguments for and against the Riemann hypothesis\nMathematical papers about the Riemann hypothesis tend to be cautiously noncommittal about its truth.\nOf authors who express an opinion, most of them, such as and, imply that they expect (or at least hope) that it is true.\nThe few authors who express serious doubt about it include, who lists some reasons for skepticism, and , who flatly states that he believes it false, that there is no evidence for it and no imaginable reason it would be true.\nThe consensus of the survey articles (,, and ) is that the evidence for it is strong but not overwhelming, so that while it is probably true there is reasonable doubt.\nSome of the arguments for and against the Riemann hypothesis are listed by, , and , and include the following:\n\n## Popular expositions\nNote: Derbyshire 2003, Rockmore 2005, Sabbagh 2003a, Sabbagh 2003b, Sautoy 2003, and Watkins 2015 are non-technical.\nEdwards 1974, Patterson 1988, Borwein/Choi/Rooney/Weirathmueller 2008, Mazur/Stein 2015, Broughan 2017, and Nahin 2021 give mathematical introductions.\nTitchmarsh 1986, Ivić 1985, and Karatsuba/Voronin 1992 are advanced [monographs](https://bliptext.com/articles/monograph).","revisionCount":1},"hasMore":false,"nextCursor":null}

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