Li's criterion and zero-free regions of -functions

✍ Scribed by Francis C.S. Brown

• Book ID: 104024493
• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 334 KB
• Volume: 111
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2004.07.016

✦ Synopsis

Let denote the Riemann zeta function, and let (s) = s (s -1) -s/2 (s/2) (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 ( ) in the first n coefficients of the Taylor expansion of 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 be any such function. For large N ∈ N, we show that the inequalities P 1 ( ), . . . , P N ( ) 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 ( ) hold. We show that the inequality P 2 ( ) implies the existence of a small zero-free region near 1, and this gives a simple condition in (1), (1), and (1), for to have no Siegel zero.