Newman's conjecture in various settings

Abstract

TextDe Bruijn and Newman introduced a deformation of the Riemann zeta function ζ(s), and found a real constant Λ which encodes the movement of the zeros of ζ(s) under the deformation. The Riemann hypothesis is equivalent to Λ≤0. Newman conjectured Λ≥0, remarking “the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so.” Previous work could only handle ζ(s) and quadratic Dirichlet L-functions, obtaining lower bounds very close to zero (−1.14541⋅10−11 for ζ(s) and −1.17⋅10−7 for quadratic Dirichlet L-functions). We generalize to automorphic L-functions and function field L-functions, and explore the limit of these techniques. If D∈Z[T] is a square-free polynomial of degree 3 and Dp the polynomial in Fp[T] obtained by reducing D modulo p, we prove the Newman constant ΛDp equals log⁡|ap(D)|2p; by Sato–Tate (if the curve is non-CM) there exists a sequence of primes such that limn→∞⁡ΛDpn=0. We end by discussing connections with random matrix theory. VideoFor a video summary of this paper, please visit http://youtu.be/8A1XZtSkp_Q. This author video is a recording of a talk given by Alan Chang at CANT on May 28, 2014.