Liʹs criterion and zero-free regions of L-functions

Let ζ denote the Riemann zeta function, and let denote the completed zeta function. A theorem of X.-J. Li states that the Riemann hypothesis is true if and only if certain inequalities in the first n coefficients of the Taylor expansion of ξ at s=1 are satisfied for all . We extend this result to a general class of functions which includes the completed Artin L-functions which satisfy Artinʹs conjecture. Now let ξ be any such function. For large , we show that the inequalities imply the existence of a certain zero-free region for ξ, and conversely, we prove that a zero-free region for ξ implies a certain number of the hold. We show that the inequality implies the existence of a small zero-free region near 1, and this gives a simple condition in ξ(1), ξ′(1), and ξ″(1), for ξ to have no Siegel zero.