TY - JOUR

T1 - Newman's conjecture in various settings

AU - Andrade, Julio

AU - Chang, Alan

AU - Miller, Steven J.

N1 - Funding Information: The first and second named authors were funded by NSF Grant DMS0850577 and Williams College , and the third named author was partially supported by NSF Grant DMS1265673 . We thank our colleagues from the 2013 Williams College SMALL REU, especially Minh-Tam Trinh, and David Geraghty, Peter Sarnak, Jeffrey Stopple and participants of the 2013 Young Mathematicians Conference at Ohio State and the Maine–Québec Number Theory Conference for many helpful conversations, and the referee for comments on an earlier draft.

PY - 2014/11

Y1 - 2014/11

N2 - Text: De 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 {dot operator} 10 -11 for ζ(s) and -1.17 {dot operator} 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 D p 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. Video: For 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.

AB - Text: De 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 {dot operator} 10 -11 for ζ(s) and -1.17 {dot operator} 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 D p 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. Video: For 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.

KW - Function fields

KW - L-functions

KW - Newman's conjecture

KW - Random matrix theory

KW - Sato-tate conjecture

KW - Zeros of the riemann zeta function

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U2 - https://doi.org/10.1016/j.jnt.2014.04.021

DO - https://doi.org/10.1016/j.jnt.2014.04.021

M3 - Article

SN - 0022-314X

VL - 144

SP - 70

EP - 91

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -