A New Proof of Newman's Conjecture and a Generalization
Abstract
Newman's conjecture (proved by Rodgers and Tao in 2018) concerns a certain family of deformations $\{\xi_t(s)\}_{t \in \mathbb{R}}$ of the Riemann xi function for which there exists an associated constant $\Lambda \in \mathbb{R}$ (called the de BruijnNewman constant) such that all the zeros of $\xi_t$ lie on the critical line if and only if $t \geq \Lambda$. The Riemann hypothesis is equivalent to the statement that $\Lambda \leq 0$, and Newman's conjecture states that $\Lambda \geq 0$. In this paper we give a new proof of Newman's conjecture which avoids many of the complications in the proof of Rodgers and Tao. Unlike the previous best methods for bounding $\Lambda$, our approach does not require any information about the zeros of the zeta function, and it can be readily be applied to a wide variety of $L$functions. In particular, we establish that any $L$function in the extended Selberg class has an associated de BruijnNewman constant and that all of these constants are nonnegative. Stated in the Riemann xi function case, our argument proceeds by showing that for every $t < 0$ the function $\xi_t$ can be approximated in terms of a Dirichlet series $\zeta_t(s)=\sum_{n=1}^{\infty}\exp(\frac{t}{4} \log^2 n)n^{s}$ whose zeros then provide infinitely many zeros of $\xi_t$ off the critical line.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.05142
 Bibcode:
 2020arXiv200505142D
 Keywords:

 Mathematics  Number Theory
 EPrint:
 29 pages, 4 figures