Shifted second moment of the Riemann zeta function and a Fourier type kernel

TL;DR

This paper computes the shifted second moment of the Riemann zeta function over extended domains and reveals its behavior as a Fourier-Cauchy type kernel, highlighting connections to orthogonal functions.

Contribution

It introduces a novel computation of the second moment with shifted arguments and demonstrates its Fourier-Cauchy kernel-like behavior on the critical line.

Findings

Second moment computed over extended domain

Behavior resembles Fourier-Cauchy kernel

Connections to orthogonal functions established

Abstract

We compute the second moment of the Riemann zeta function for shifted arguments over a domain that extends the ones in the literature. We use the Riemann-Siegel formula for the error term in the approximate functional equation and take the products of all the terms into account. We also show that, as a function of imaginary shifts on the critical line, the the second moment behaves like a Fourier-Cauchy type kernel on a class of functions. This is reminiscent of orthogonal functions.