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 ∈ R, define the entire functionwhere Φ is the super-exponentially decaying functionThis 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 de Bruijn-Newman constant) such that the zeroes of H t are all real precisely when t ≥ Λ. The Riemann hypothesis is equivalent to the assertion Λ ≤ 0; recently, Rodgers and Tao established the matching lower bound Λ ≥ 0. Ki, Kim and Lee established the upper bound Λ < 1 2 . In this paper we establish several effective estimates on H t (x + iy) for t ≥ 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 ≤ 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 → ∞.

Will a physicist prove the Riemann hypothesis?

“…At present the Polymath Project (collaborations of many people to solve mathematical problems) is led by Tao on the topic of obtaining new upper bounds on the de Bruijn-Newman constant Λ. They showed that Λ < 0.22, see [58], the present bounds on the de Bruijn-Newman constant are 0 Λ < 0.22. Related to this problems is a recent paper [59].…”

Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman

“…The earliest was de Bruijn's Λ ≤ 1/2 [45] in his original paper; an improvement to a strict inequality Λ < 1/2 was made in 2009 by Ki, Kim, and Lee [147]. An improved upper bound of 0.22 was very recently obtained by Tao and collaborators, as posted in [241]. In addition, a survey article by Chuck and Wei Wu on de Bruijn-Newman type constants has just been published [227].…”

Entire solutions for the heat equation

“…for all but countably many t ∈ C. The above equations were first derived by Csordas, Smith, and Varga [5] for the case where t is restricted in a real semiaxis (see [16,18]), since one must use Theorem 4.1 of the present paper in order to analytically extend (1.24) to t ∈ C. The equations (1.24) are a kind of " characteristics" for the heat equation and they also remind the equations which arise in the solution of the inverse spectral problem for the Hill operator (see [20]).…”