A Parabolic Cylinder Function in the Riemann-Siegel Integral Formula

TL;DR

This paper demonstrates how the integrals in the Riemann-Siegel formula can be expressed using the parabolic cylinder function, providing new integral representations.

Contribution

It introduces a novel transformation of the Riemann-Siegel integral formula involving the parabolic cylinder function as the kernel.

Findings

Integral representations contain the parabolic cylinder function $U(a,z)$

Transformation offers new analytical tools for the Riemann-Siegel formula

Potential applications in analytic number theory and zeta function analysis

Abstract

We show that the two integrals in the Riemann-Siegel integral formula can be transformed into integral representations that contain the parabolic cylinder function $U(a,z)$ as kernel function.