On new representations for values of Riemann zeta function at odd points and related numbers

Abstract

Let $\zeta(s)$ and $\beta(s)$ be the Riemann zeta function and Dirichlet beta function. In this work, for some linear combinations of the numbers
$\zeta(2n+1)$ and $\beta(2n)$, we obtain new representations by the series, the general term of which involves the logarithms. This is done by the methods of spectral theory of ordinary differential operators generated in the Hilbert space $\mathcal{L}^2[0,\pi]$ by the expression $l[y]=-y''-a^2y$ and the Dirichlet boundary condition, where $a$ is a parameter. These representations in particular imply the known and new representations for these linear combinations as the sums of some sufficiently fast converging series, the general term of which involves $\zeta(2n)$. The obtained results are applied to various representations of Catalan constant $\beta(2)$ and Apéry constants $\zeta(3)$.