Functiones et Approximatio Commentarii Mathematici

Notes on low discriminants and the generalized Newman conjecture

Jeffrey Stopple

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Abstract

Generalizing work of Polya, de Bruijn and Newman, we allow the backward heat equation to deform the zeros of quadratic Dirichlet $L$-functions. There is a real constant $\Lambda_{Kr}$ (generalizing the de Bruijn-Newman constant $\Lambda$) such that for time $t\ge\Lambda_{Kr}$ all such $L$-functions have all their zeros on the critical line; for time $t<\Lambda_{Kr}$ there exist zeros off the line. Under GRH, $\Lambda_{Kr}\le 0$; we make the complementary conjecture $0\le \Lambda_{Kr}$. Following the work of Csordas \emph{et. al}. on Lehmer pairs of Riemann zeros, we use low-lying zeros of quadratic Dirichlet $L$-functions to show that $-1.13\cdot 10^{-7}<\Lambda_{Kr}$. In the last section we develop a precise definition of a Low discriminant which is motivated by considerations of random matrix theory. The existence of infinitely many Low discriminants would imply $0\le \Lambda_{Kr}$.

Article information

Source
Funct. Approx. Comment. Math., Volume 51, Number 1 (2014), 23-41.

Dates
First available in Project Euclid: 24 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.facm/1411564614

Digital Object Identifier
doi:10.7169/facm/2014.51.1.2

Mathematical Reviews number (MathSciNet)
MR3263068

Zentralblatt MATH identifier
1357.11078

Subjects
Primary: 11M20: Real zeros of $L(s, \chi)$; results on $L(1, \chi)$
Secondary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 11M50: Relations with random matrices 11Y35: Analytic computations 11Y60: Evaluation of constants

Keywords
generalized Riemann hypothesis de Bruijn-Newman constant backward heat equation Lehmer pair Low discriminant random matrix theory

Citation

Stopple, Jeffrey. Notes on low discriminants and the generalized Newman conjecture. Funct. Approx. Comment. Math. 51 (2014), no. 1, 23--41. doi:10.7169/facm/2014.51.1.2. https://projecteuclid.org/euclid.facm/1411564614


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