Effective approximation of heat flow evolution of the Riemann $\xi$ function, and a new upper bound for the de Bruijn-Newman constant

Abstract

For each $t\in \mathbf{R}$, define the entire function $$H_t(z):={\int }_0^{\infty }{e}^{t{u}^2}\Phi (u)\cos (zu)du$$ where $\Phi$ is the super-exponentially decaying function $$\Phi (u):=\sum _{n=1}^{\infty }(2{\pi }^2{n}^4{e}^{9u}-3\pi n^2{e}^{5u})\exp (-\pi n^2{e}^{4u}).$$ This is essentially the heat flow evolution of the Riemann $\xi$ function. From the work of de Bruijn and Newman, there exists a finite constant $\Lambda$ (the \emph{de Bruijn-Newman constant}) such that the zeroes of $H_t$ are all real precisely when $t\ge \Lambda$. The Riemann hypothesis is equivalent to the assertion $\Lambda \le 0$; recently, Rodgers and Tao established the matching lower bound $\Lambda \ge 0$. Ki, Kim and Lee established the upper bound $\Lambda <\frac{1}{2}$. In this paper we establish several effective estimates on $H_t(x+iy)$ for $t\ge 0$, including some that are accurate for small or medium values of $x$. By combining these estimates with numerical computations, we are able to obtain a new upper bound $\Lambda \le 0.22$ unconditionally, as well as improvements conditional on further numerical verification of the Riemann hypothesis. We also obtain some new estimates controlling the asymptotic behavior of zeroes of $H_t(x+iy)$ as $x\to \infty$.

68 pages, 21 figures. To appear, Research in the Mathematical Sciences. This is the final version, incorporating referee comments and corrections