Some Definite Integrals Associated with the Riemann Zeta Function

Abstract

The authors aim at deriving a family of series representations for $\zeta (2n+1)(n\in \mathbb{N})$ by evaluating certain trigonometric integrals in several different ways. They also show how the results presented in this paper relate to those that were obtained in other works. Finally, some illustrative computational examples, using $Mathematica$ (Version 4.0) for Linux, are considered.