Li's criterion and zero-free regions of [formula omitted]-functions

Abstract

Let ζ denote the Riemann zeta function, and let ξ ( s ) = s ( s - 1 ) π - s / 2 Γ ( s / 2 ) ζ ( s ) denote the completed zeta function. A theorem of X.-J. Li states that the Riemann hypothesis is true if and only if certain inequalities P n ( ξ ) in the first n coefficients of the Taylor expansion of ξ at s = 1 are satisfied for all n ∈ N . We extend this result to a general class of functions which includes the completed Artin L-functions which satisfy Artin's conjecture. Now let ξ be any such function. For large N ∈ N , we show that the inequalities P 1 ( ξ ) , … , P N ( ξ ) 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 P n ( ξ ) hold. We show that the inequality P 2 ( ξ ) 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.