In number theory, Li's criterion is a particular statement about the positivity of a certain sequence that is equivalent to the Riemann hypothesis. The criterion is named after Xian-Jin Li, who presented it in 1997. In 1999, Enrico Bombieri and Jeffrey C. Lagarias provided a generalization, showing that Li's positivity condition applies to any collection of points that lie on the Re(s) = 1/2 axis.

Definition

The [[Riemann Xi function|Riemann ξ function]] is given by where ζ is the Riemann zeta function. Consider the sequence Li's criterion is then the statement that The numbers \lambda_n (sometimes defined with a slightly different normalization) are called Keiper-Li coefficients or Li coefficients. They may also be expressed in terms of the non-trivial zeros of the Riemann zeta function: where the sum extends over ρ, the non-trivial zeros of the zeta function. This conditionally convergent sum should be understood in the sense that is usually used in number theory, namely, that (Re(s) and Im(s) denote the real and imaginary parts of s, respectively.) The positivity of \lambda_n has been verified up to n = 10^5 by direct computation.

Proof

Note that. Then, starting with an entire function, let. \phi vanishes when. Hence, is holomorphic on the unit disk |z| < 1 iff. Write the Taylor series. Since we have so that Finally, if each zero \rho comes paired with its complex conjugate \bar{\rho}, then we may combine terms to get The condition then becomes equivalent to. The right-hand side of is obviously nonnegative when both n \ge 0 and. Conversely, ordering the \rho by, we see that the largest term dominates the sum as , and hence c_n becomes negative sometimes.

A generalization

Bombieri and Lagarias demonstrate that a similar criterion holds for any collection of complex numbers, and is thus not restricted to the Riemann hypothesis. More precisely, let R = {ρ} be any collection of complex numbers ρ, not containing ρ = 1, which satisfies Then one may make several equivalent statements about such a set. One such statement is the following: One may make a more interesting statement, if the set R obeys a certain functional equation under the replacement s ↦ 1 − s. Namely, if, whenever ρ is in R, then both the complex conjugate and 1-\rho are in R, then Li's criterion can be stated as: Bombieri and Lagarias also show that Li's criterion follows from Weil's criterion for the Riemann hypothesis.