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 $$ H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du$$ 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 $t \geq \Lambda$. The Riemann hypothesis is equivalent to the assertion $\Lambda \leq 0$; recently, Rodgers and Tao established the matching lower bound $\Lambda \geq 0$. 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 $t \geq 0$, 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 $\Lambda \leq 0.22$ 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 $x \to \infty$.