Representations of $\zeta (2n + 1)$ and Related Numbers in the Form of Definite Integrals and Rapidly Convergent Series

Abstract

Let $\zeta (s)$ and $\beta (s)$ be the Riemann zeta function and the Dirichlet beta function. The formulas for calculating the values of $\zeta (2m)$ and $\beta (2m - 1)$ ($m = 1,\;2,\; \ldots$) are classical and well known. Our aim is to represent $\zeta (2m + 1)$, $\beta (2m)$, and related numbers in the form of definite integrals of elementary functions and rapidly converging numerical series containing $\zeta (2m)$. By applying the method of this work, on the one hand, both classical formulas and ones relatively recently obtained by others researchers are proved in a uniform manner, and on the other hand, numerous new results are derived.