Special values of the Lerch zeta function and the evaluation of certain integrals

Abstract:

The Lerch zeta function $\Phi (x,a,s)$ is defined by the series \[ \Phi (x,a,s) = \sum \limits _{n = 0}^\infty {\frac {{{e^{2n\pi ix}}}} {{{{(n + a)}^s}}}} ,\] where $x$ is real, $0 < a \leqslant 1$, and $\sigma = \operatorname {Re} (s) > 1$ if $x$ is an integer and $\sigma > 0$ otherwise. In this paper we study the function $J\left ( {s,a} \right ) = \Phi (\tfrac {1} {2},a,s)$. We use its integral representation \[ J\left ( {s,a} \right ) = \frac {{{a^{ - s}}}} {2} + 2\int _0^\infty {{{({a^2} + {y^2})}^{ - s/2}}\sin \left ( {s {{\tan }^{ - 1}}\frac {y} {a}} \right )} \frac {{{e^{\pi y}}dy}} {{{e^{2\pi y}} - 1}}\] to obtain the values of certain definite integrals; for example, we show that \[ \begin {gathered} \int _0^\infty {\frac {{\cosh x\log x}} {{\cosh 2x - \cos 2\pi a}}} dx \hfill \\ \qquad = \frac {\pi } {{2\sin \pi a}}\left \{ {\log \frac {{\Gamma ((1 + a)/2)}} {{\Gamma (a/2)}} + \frac {1} {2}\log \left ( {2\pi \cot \frac {{\pi a}} {2}} \right )} \right \},\qquad 0 < a < 1. \hfill \\ \end {gathered} \]