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

Abstract

For each $t \in \mathbf{R}$, define the entire function $$ H_t(z) := \int_0^\infty e^{tu^2} Φ(u) \cos(zu)\ du$$ where $Φ$ is the super-exponentially decaying function $$ Φ(u) := \sum_{n=1}^\infty (2π^2 n^4 e^{9u} - 3πn^2 e^{5u} ) \exp(-πn^2 e^{4u} ).$$ This is essentially the heat flow evolution of the Riemann $ξ$ function. From the work of de Bruijn and Newman, there exists a finite constant $Λ$ (the \emph{de Bruijn-Newman constant}) such that the zeroes of $H_t$ are all real precisely when $t \geq Λ$. The Riemann hypothesis is equivalent to the assertion $Λ\leq 0$; recently, Rodgers and Tao established the matching lower bound $Λ\geq 0$. Ki, Kim and Lee established the upper bound $Λ< \frac{1}{2}$. 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 $Λ\leq 0.22$ 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 $x \to \infty$.