Let xi denote the Riemann zeta function, and let xi(s) = s (s - 1)pi(-s/2)Gamma(s/2)zeta(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(xi) in the first n coefficients of the Taylor expansion of xi at s = 1 are satisfied for all n epsilon 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 be any such function. For large N is an element of N, we show that the inequalities P-1(xi), . . ., P-N(xi) imply the existence of a certain zero-free region for, and conversely, we prove that a zero-free region for implies a certain number of the P-n(xi) hold. We show that the inequality P-2(xi) implies the existence of a small zero-free region near 1, and this gives a simple condition in xi(1), xi'(1), xi"(1) and for xi to have no Siegel zero. (c) 2004 Elsevier Inc. All rights reserved.
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