Closed-form evaluations of definite integrals and associated infinite series involving the Riemann zeta function

Abstract

A class of definite integrals are evaluated using the Cauchy residue theorem. These evaluations are then used to obtain closed-form expressions for several infinite series associated with the Riemann zeta function. A number of interesting (known or new) special cases and consequences of the main results are also considered.

Keywords:

• Riemann zeta function
• Hurwitz zeta function
• Gamma functions
• Multiple gamma functions
• Cauchy residue theorem
• Hurwitz–Lerch zeta function
• Generalized zeta function

Acknowledgements

The present investigation was supported by the Dongguk University Research Fund and the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.