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.