On certain Fourier expansions for the Riemann zeta function

TL;DR

This paper develops new Fourier expansions for the Riemann zeta and related L-functions, introduces series involving Whittaker functions, and presents a novel expansion for the Riemann xi function through Mellin transform techniques.

Contribution

It provides new Fourier and series expansions for the Riemann zeta, its reciprocal, and the xi function, extending previous work with novel integral and series representations.

Findings

Fourier expansions for certain L-functions established

Series involving Whittaker functions derived for coefficients

New expansion for the Riemann xi function using Mellin transforms

Abstract

We build on a recent paper on Fourier expansions for the Riemann zeta function. We establish Fourier expansions for certain $L$-functions, and offer series representations involving the Whittaker function $W_{\gamma,\mu}(z)$ for the coefficients. Fourier expansions for the reciprocal of the Riemann zeta function are also stated. A new expansion for the Riemann xi function is presented in the third section by constructing an integral formula using Mellin transforms for its Fourier coefficients.