{"history":[],"article":{"id":"07a79274-7ff0-4882-92a6-f5ad7225e50b","title":"Hurwitz zeta function","slug":"hurwitz-zeta-function","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **Hurwitz zeta function** is one of the many [zeta functions](https://bliptext.com/articles/zeta-function).\nIt is formally defined for [complex](https://bliptext.com/articles/complex-number) variables s with\nRe(s) > 1\nand\na ≠ 0, −1, −2, …\nby\nThis series is [absolutely convergent](https://bliptext.com/articles/absolutely-convergent) for the given values of s and a and can be extended to a [meromorphic function](https://bliptext.com/articles/meromorphic-function) defined for all\ns ≠ 1\n. The [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) is\nζ(s,1)\n. The Hurwitz zeta function is named after [Adolf Hurwitz](https://bliptext.com/articles/adolf-hurwitz), who introduced it in 1882.\n[[File:Hurwitza1ov3v2.png|right|thumb|Hurwitz zeta function corresponding to\na = 1/3\n. It is generated as a [Matplotlib](https://bliptext.com/articles/matplotlib) plot using a version of the [Domain coloring](https://bliptext.com/articles/domain-coloring) method.\n]] [[File:Hurwitza24ov25v2.png|right|thumb|Hurwitz zeta function corresponding to\na = 24/25\n.]] [[File:HurwitzofAz3p4j.png|right|thumb| Hurwitz zeta function as a function of a with\ns = 3 + 4i\n.]]\n\n# Integral representation\nThe Hurwitz zeta function has an integral representation\nfor and (This integral can be viewed as a [Mellin transform](https://bliptext.com/articles/mellin-transform).) The formula can be obtained, roughly, by writing\nand then interchanging the sum and integral.\nThe integral representation above can be converted to a [contour integral](https://bliptext.com/articles/contour-integral) representation\nwhere C is a [Hankel contour](https://bliptext.com/articles/hankel-contour) counterclockwise around the positive real axis, and the [principal branch](https://bliptext.com/articles/principal-branch) is used for the [complex exponentiation](https://bliptext.com/articles/complex-exponentiation) (-z)^{s-1}.\nUnlike the previous integral, this integral is valid for all s, and indeed is an [entire function](https://bliptext.com/articles/entire-function) of s.\nThe contour integral representation provides an [analytic continuation](https://bliptext.com/articles/analytic-continuation) of \\zeta(s,a) to all s \\ne 1.\nAt s = 1, it has a [simple pole](https://bliptext.com/articles/simple-pole) with [residue](https://bliptext.com/articles/residue-complex-analysis) 1.\n\n# Hurwitz's formula\nThe Hurwitz zeta function satisfies an identity which generalizes the [functional equation of the Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function):\nvalid for Re(s) > 1 and 0 < a ≤ 1.\nThe Riemann zeta functional equation is the special case a = 1:\nHurwitz's formula can also be expressed as (for Re(s) < 0 and 0 < a ≤ 1).\nHurwitz's formula has a variety of different proofs.\nOne proof uses the contour integration representation along with the [residue theorem](https://bliptext.com/articles/residue-theorem).\nA second proof uses a [theta function](https://bliptext.com/articles/theta-function) identity, or equivalently [Poisson summation](https://bliptext.com/articles/poisson-summation).\nThese proofs are analogous to the two proofs of the functional equation for the Riemann zeta function in [Riemann's 1859 paper](https://bliptext.com/articles/on-the-number-of-primes-less-than-a-given-magnitude).\nAnother proof of the Hurwitz formula uses [Euler–Maclaurin summation](https://bliptext.com/articles/euler-maclaurin-summation) to express the Hurwitz zeta function as an integral\n(−1 < Re(s) < 0 and 0 < a ≤ 1) and then expanding the numerator as a [Fourier series](https://bliptext.com/articles/fourier-series).\n\n## Functional equation for rational a\nWhen a is a rational number, Hurwitz's formula leads to the following [functional equation](https://bliptext.com/articles/functional-equation): For integers ,\nholds for all values of s.\nThis functional equation can be written as another equivalent form:\n.\n\n# Some finite sums\nClosely related to the functional equation are the following finite sums, some of which may be evaluated in a closed form\nwhere m is positive integer greater than 2 and s is complex, see e.g. Appendix B in.\n\n# Series representation\nA convergent [Newton series](https://bliptext.com/articles/newton-series) representation defined for (real) a > 0 and any complex s ≠ 1 was given by [Helmut Hasse](https://bliptext.com/articles/helmut-hasse) in 1930:\nThis series converges uniformly on [compact subsets](https://bliptext.com/articles/compact-subset) of the s-plane to an [entire function](https://bliptext.com/articles/entire-function).\nThe inner sum may be understood to be the nth [forward difference](https://bliptext.com/articles/forward-difference) of a^{1-s}; that is,\nwhere Δ is the [forward difference operator](https://bliptext.com/articles/forward-difference-operator).\nThus, one may write:\n\n# Taylor series\nThe partial derivative of the zeta in the second argument is a [shift](https://bliptext.com/articles/sheffer-sequence):\nThus, the [Taylor series](https://bliptext.com/articles/taylor-series) can be written as:\nAlternatively,\nwith |q| < 1.\nClosely related is the **Stark–Keiper** formula:\nwhich holds for integer N and arbitrary s.\nSee also [Faulhaber's formula](https://bliptext.com/articles/faulhaber-s-formula) for a similar relation on finite sums of powers of integers.\n\n# Laurent series\nThe [Laurent series](https://bliptext.com/articles/laurent-series) expansion can be used to define generalized [Stieltjes constants](https://bliptext.com/articles/stieltjes-constants) that occur in the series\nIn particular, the constant term is given by where \\Gamma is the [gamma function](https://bliptext.com/articles/gamma-function) and is the di[gamma function](https://bliptext.com/articles/gamma-function).\nAs a special case,.\n\n# Discrete Fourier transform\nThe [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) of the Hurwitz zeta function with respect to the order s is the [Legendre chi function](https://bliptext.com/articles/legendre-chi-function).\n\n# Particular values\n\n## Negative integers\nThe values of ζ(s, a) at s = 0, −1, −2, ...\nare related to the [Bernoulli polynomials](https://bliptext.com/articles/bernoulli-polynomials):\nFor example, the n=0 case gives\n\n## s-derivative\nThe [partial derivative](https://bliptext.com/articles/partial-derivative) with respect to s at s = 0 is related to the gamma function:\nIn particular, The formula is due to [Lerch](https://bliptext.com/articles/mathias-lerch).\n\n# Relation to Jacobi theta function\nIf is the Jacobi [theta function](https://bliptext.com/articles/theta-function), then\nholds for \\Re s > 0 and z complex, but not an integer.\nFor z=n an integer, this simplifies to\nwhere ζ here is the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function).\nNote that this latter form is the [functional equation](https://bliptext.com/articles/functional-equation) for the Riemann zeta function, as originally given by Riemann.\nThe distinction based on z being an integer or not accounts for the fact that the Jacobi theta function converges to the periodic [delta function](https://bliptext.com/articles/dirac-delta-function), or [Dirac comb](https://bliptext.com/articles/dirac-comb) in z as.\n\n# Relation to Dirichlet L-functions\nAt rational arguments the Hurwitz zeta function may be expressed as a linear combination of [Dirichlet L-functions](https://bliptext.com/articles/dirichlet-l-function) and vice versa: The Hurwitz zeta function coincides with [Riemann's zeta function](https://bliptext.com/articles/riemann-zeta-function) ζ(s) when a = 1, when a = 1/2 it is equal to (2s−1)ζ(s), and if a = n/k with k > 2, (n,k) > 1 and 0 < n < k, then\nthe sum running over all [Dirichlet characters](https://bliptext.com/articles/dirichlet-character) mod k.\nIn the opposite direction we have the linear combination\nThere is also the [multiplication theorem](https://bliptext.com/articles/multiplication-theorem)\nof which a useful generalization is the distribution relation\n(This last form is valid whenever q a natural number and 1 − qa is not.)\n\n# Zeros\nIf a=1 the Hurwitz zeta function reduces to the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) itself; if a=1/2 it reduces to the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) multiplied by a simple function of the complex argument s (vide supra), leading in each case to the difficult study of the zeros of Riemann's zeta function.\nIn particular, there will be no zeros with real part greater than or equal to 1.\nHowever, if 0 1\nand\na ≠ 0, −1, −2, …\nby\nThis series is [absolutely convergent](https://bliptext.com/articles/absolutely-convergent) for the given values of s and a and can be extended to a [meromorphic function](https://bliptext.com/articles/meromorphic-function) defined for all\ns ≠ 1\n. The [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) is\nζ(s,1)\n. The Hurwitz zeta function is named after [Adolf Hurwitz](https://bliptext.com/articles/adolf-hurwitz), who introduced it in 1882.\n[[File:Hurwitza1ov3v2.png|right|thumb|Hurwitz zeta function corresponding to\na = 1/3\n. It is generated as a [Matplotlib](https://bliptext.com/articles/matplotlib) plot using a version of the [Domain coloring](https://bliptext.com/articles/domain-coloring) method.\n]] [[File:Hurwitza24ov25v2.png|right|thumb|Hurwitz zeta function corresponding to\na = 24/25\n.]] [[File:HurwitzofAz3p4j.png|right|thumb| Hurwitz zeta function as a function of a with\ns = 3 + 4i\n.]]\n\n# Integral representation\nThe Hurwitz zeta function has an integral representation\nfor and (This integral can be viewed as a [Mellin transform](https://bliptext.com/articles/mellin-transform).) The formula can be obtained, roughly, by writing\nand then interchanging the sum and integral.\nThe integral representation above can be converted to a [contour integral](https://bliptext.com/articles/contour-integral) representation\nwhere C is a [Hankel contour](https://bliptext.com/articles/hankel-contour) counterclockwise around the positive real axis, and the [principal branch](https://bliptext.com/articles/principal-branch) is used for the [complex exponentiation](https://bliptext.com/articles/complex-exponentiation) (-z)^{s-1}.\nUnlike the previous integral, this integral is valid for all s, and indeed is an [entire function](https://bliptext.com/articles/entire-function) of s.\nThe contour integral representation provides an [analytic continuation](https://bliptext.com/articles/analytic-continuation) of \\zeta(s,a) to all s \\ne 1.\nAt s = 1, it has a [simple pole](https://bliptext.com/articles/simple-pole) with [residue](https://bliptext.com/articles/residue-complex-analysis) 1.\n\n# Hurwitz's formula\nThe Hurwitz zeta function satisfies an identity which generalizes the [functional equation of the Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function):\nvalid for Re(s) > 1 and 0 0 and any complex s ≠ 1 was given by [Helmut Hasse](https://bliptext.com/articles/helmut-hasse) in 1930:\nThis series converges uniformly on [compact subsets](https://bliptext.com/articles/compact-subset) of the s-plane to an [entire function](https://bliptext.com/articles/entire-function).\nThe inner sum may be understood to be the nth [forward difference](https://bliptext.com/articles/forward-difference) of a^{1-s}; that is,\nwhere Δ is the [forward difference operator](https://bliptext.com/articles/forward-difference-operator).\nThus, one may write:\n\n# Taylor series\nThe partial derivative of the zeta in the second argument is a [shift](https://bliptext.com/articles/sheffer-sequence):\nThus, the [Taylor series](https://bliptext.com/articles/taylor-series) can be written as:\nAlternatively,\nwith |q| 0 and z complex, but not an integer.\nFor z=n an integer, this simplifies to\nwhere ζ here is the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function).\nNote that this latter form is the [functional equation](https://bliptext.com/articles/functional-equation) for the Riemann zeta function, as originally given by Riemann.\nThe distinction based on z being an integer or not accounts for the fact that the Jacobi theta function converges to the periodic [delta function](https://bliptext.com/articles/dirac-delta-function), or [Dirac comb](https://bliptext.com/articles/dirac-comb) in z as.\n\n# Relation to Dirichlet L-functions\nAt rational arguments the Hurwitz zeta function may be expressed as a linear combination of [Dirichlet L-functions](https://bliptext.com/articles/dirichlet-l-function) and vice versa: The Hurwitz zeta function coincides with [Riemann's zeta function](https://bliptext.com/articles/riemann-zeta-function) ζ(s) when a = 1, when a = 1/2 it is equal to (2s−1)ζ(s), and if a = n/k with k > 2, (n,k) > 1 and 0 < n < k, then\nthe sum running over all [Dirichlet characters](https://bliptext.com/articles/dirichlet-character) mod k.\nIn the opposite direction we have the linear combination\nThere is also the [multiplication theorem](https://bliptext.com/articles/multiplication-theorem)\nof which a useful generalization is the distribution relation\n(This last form is valid whenever q a natural number and 1 − qa is not.)\n\n# Zeros\nIf a=1 the Hurwitz zeta function reduces to the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) itself; if a=1/2 it reduces to the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) multiplied by a simple function of the complex argument s (vide supra), leading in each case to the difficult study of the zeros of Riemann's zeta function.\nIn particular, there will be no zeros with real part greater than or equal to 1.\nHowever, if 0

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