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

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

2011-01-29

arxiv.org/abs/1101.5722v1

Physics

Mathematical Physics

30 pages, no figures

Scientific paper

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.

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