Constants of de Bruijn–Newman type in analytic number theory and statistical physics

Abstract

One formulation in 1859 of the Riemann Hypothesis (RH) was that the Fourier transform $H_f(z)$ of $f$ for $ z \in \mathbb{C}$ has only real zeros when $f(t)$ is a specific function $\Phi (t)$. P\'{o}lya's 1920s approach to RH extended $H_f$ to $H_{f,\lambda}$, the Fourier transform of $e^{\lambda t^2} f(t)$. We review developments of this approach to RH and related ones in statistical physics where $f(t)$ is replaced by a measure $d \rho (t)$. P\'{o}lya's work together with 1950 and 1976 results of de Bruijn and Newman, respectively, imply the existence of a finite constant $\Lambda_{DN} = \Lambda_{DN} (\Phi)$ in $(-\infty, 1/2]$ such that $H_{\Phi,\lambda}$ has only real zeros if and only if $\lambda \geq \Lambda_{DN}$; RH is then equivalent to $\Lambda_{DN} \leq 0$. Recent developments include the Rodgers and Tao proof of the 1976 conjecture that $\Lambda_{DN} \geq 0$ (that RH, if true, is only barely so) and the Polymath 15 project improving the $1/2$ upper bound to about $0.22$. We also present examples of $\rho$'s with differing $H_{\rho,\lambda}$ and $\Lambda_{DN} (\rho)$ behaviors; some of these are new and based on a recent weak convergence theorem of the authors.