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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Kenneth S. Williams and Nan Yue Zhang
Journal: Proc. Amer. Math. Soc. 119 (1993), 35-49
MSC: Primary 11M35; Secondary 11M06
MathSciNet review: 1172963
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Abstract: The Lerch zeta function $ \Phi (x,a,s)$ is defined by the series

$\displaystyle \Phi (x,a,s) = \sum\limits_{n = 0}^\infty {\frac{{{e^{2n\pi ix}}}} {{{{(n + a)}^s}}}} ,$

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

$\displaystyle J\left( {s,a} \right) = \frac{{{a^{ - s}}}} {2} + 2\int_0^\infty ... ...tan }^{ - 1}}\frac{y} {a}} \right)} \frac{{{e^{\pi y}}dy}} {{{e^{2\pi y}} - 1}}$

to obtain the values of certain definite integrals; for example, we show that

\begin{displaymath}\begin{gathered}\int_0^\infty {\frac{{\cosh x\log x}} {{\cosh... ...} \right)} \right\},\qquad 0 < a < 1. \hfill \\ \end{gathered} \end{displaymath}


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1172963-7
PII: S 0002-9939(1993)1172963-7
Keywords: Lerch zeta function, Hurwitz zeta function, integral representation, recurrence relations
Article copyright: © Copyright 1993 American Mathematical Society