THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE

Abstract

For each$t\in \mathbb{R}$, we define the entire function$$\begin{eqnarray}H_{t}(z):=\int _{0}^{\infty }e^{tu^{2}}\unicode[STIX]{x1D6F7}(u)\cos (zu)\,du,\end{eqnarray}$$where$\unicode[STIX]{x1D6F7}$is the super-exponentially decaying function$$\begin{eqnarray}\unicode[STIX]{x1D6F7}(u):=\mathop{\sum }_{n=1}^{\infty }(2\unicode[STIX]{x1D70B}^{2}n^{4}e^{9u}-3\unicode[STIX]{x1D70B}n^{2}e^{5u})\exp (-\unicode[STIX]{x1D70B}n^{2}e^{4u}).\end{eqnarray}$$Newman showed that there exists a finite constant$\unicode[STIX]{x1D6EC}$(thede Bruijn–Newman constant) such that the zeros of$H_{t}$are all real precisely when$t\geqslant \unicode[STIX]{x1D6EC}$. The Riemann hypothesis is equivalent to the assertion$\unicode[STIX]{x1D6EC}\leqslant 0$, and Newman conjectured the complementary bound$\unicode[STIX]{x1D6EC}\geqslant 0$. In this paper, we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that$\unicode[STIX]{x1D6EC}<0$and then analyzing the dynamics of zeros of$H_{t}$(building on the work of Csordas, Smith and Varga) to obtain increasingly strong control on the zeros of$H_{t}$in the range$\unicode[STIX]{x1D6EC}<t\leqslant 0$, until one establishes that the zeros of$H_{0}$are in local equilibrium, in the sense that they locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeros of the Riemann zeta function, such as the pair correlation estimates of Montgomery.