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 $$t \in \mathbb {R}$$ , define the entire function $$\begin{aligned} H_t(z){:=}\,\int _0^\infty e^{tu^2} \varPhi (u) \cos (zu)\ \mathrm{d}u, \end{aligned}$$ where $$\varPhi $$ is the super-exponentially decaying function $$\begin{aligned} \varPhi (u){:=}\,\sum _{n=1}^\infty (2\pi ^2 n^4 e^{9u} - 3\pi n^2 e^{5u} ) \exp (-\pi n^2 e^{4u}). \end{aligned}$$ 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 $$\varLambda $$ (the de Bruijn–Newman constant) such that the zeroes of $$H_t$$ are all real precisely when $$t \ge \varLambda $$ . The Riemann hypothesis is equivalent to the assertion $$\varLambda \le 0$$ ; recently, Rodgers and Tao established the matching lower bound $$\varLambda \ge 0$$ . Ki, and Kim and Lee established the upper bound $$\varLambda < \frac{1}{2}$$ . 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 $$\varLambda \le 0.22$$ 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 $$x \rightarrow \infty $$ .