Evaluation of some second moment and other integrals for the Riemann, Hurwitz, and Lerch zeta functions

TL;DR

This paper evaluates second moments and integrals involving the Riemann, Hurwitz, and Lerch zeta functions, providing new formulas, corollaries, and asymptotic results relevant to analytic number theory.

Contribution

It presents new integral evaluations and corollaries for the zeta functions, extending known results and exploring asymptotic behaviors and alternative approaches.

Findings

Explicit second moment integral formulas for zeta functions

Corollaries including known special cases

Asymptotic forms of fractional part integrals

Abstract

Several second moment and other integral evaluations for the Riemann zeta function $\zeta(s)$, Hurwitz zeta function $\zeta(s,a)$, and Lerch zeta function $\Phi(z,s,a)$ are presented. Additional corollaries that are obtained include previously known special cases for the Riemann zeta function $\zeta(s)=\zeta(s,1)=\Phi(1,s,1)$. An example special case is: $$\int_R {{|\zeta(1/2+it)|^2} \over {t^2+1/4}}dt=2\pi[\ln(2\pi)-\gamma],$$ with $\gamma$ the Euler constant. The asymptotic forms of certain fractional part integrals, with and without logarithmic factors in the integrand, are presented. Extensions and other approaches are mentioned.