Representations of $\zeta (2n + 1)$ and Related Numbers in the Form
of Definite Integrals and Rapidly Convergent Series
K. A. Mirzoeva,* and T. A. Safonovab,**
a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, 119991 Russia
b Lomonosov Northern Arctic Federal University, Arkhangelsk, 163002 Russia
Correspondence to: *e-mail: mirzoev.karahan@mail.ru
Correspondence to: **e-mail: t.Safonova@narfu.ru
Received 14 July, 2020
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.
Keywords: integral representation of series sums, values of the Riemann zeta function at odd points, values of the Dirichlet beta function at even points, Catalan’s and Apéry's constants
DOI: 10.1134/S1064562420050361