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 $\Phi (t)$. Pólya’s 1920s approach to the 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 the RH and related ones in statistical physics where $f(t)$ is replaced by a measure $d \rho (t)$. Pólya’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 $\lambda \geq \Lambda _{DN}$; the 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 the 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.