A proof of Newman’s conjecture for the extended Selberg class
Volume 201 / 2021
Abstract
Newman’s conjecture (proved by Rodgers and Tao in 2018) concerns a certain family $\{\xi _t(s)\}_{t \in \mathbb R}$ of deformations of the Riemann xi function for which there exists an associated constant $\Lambda \in \mathbb R$ (called the de Bruijn-Newman 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’s. 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 applied to a wide variety of $L$-functions. In particular, we establish that any $L$-func\-tion in the extended Selberg class has an associated de Bruijn–Newman constant and that all of these constants are nonnegative.
Stated in the Riemann xi function case, our argument proceeds by showing that for every $t \lt 0$ the function $\xi _t$ can be approximated in terms of a Dirichlet series $\zeta _t(s)=\sum _{n=1}^{\infty }\exp \left (\frac {t}{4} \log ^2 n\right )n^{-s}$ whose zeros then provide infinitely many zeros of $\xi _t$ off the critical line.
