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

  • D. H. J. PolymathEmail author


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 \(x \rightarrow \infty \).



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