A simpler proof of a Katsurada's theorem and rapidly converging series for $\zeta{(2n+1)}$ and $\beta{(2n)}$

TL;DR

This paper presents a simplified proof of Katsurada's theorem and introduces rapidly converging series for odd zeta values and even Dirichlet beta values, improving computational efficiency for constants like ζ(3) and G.

Contribution

It provides a simpler proof of Katsurada's theorem and derives faster converging series for ζ(3) and G, enhancing computational methods for these constants.

Findings

Series for ζ(3) converge faster than Apéry's series.

Series for G converge faster than Ramanujan's series.

The paper offers a more direct proof of Katsurada's theorem.

Abstract

In a recent work on Euler-type formulae for even Dirichlet beta values, i.e. $\beta{(2n)}$, I have derived an exact closed-form expression for a class of zeta series. From this result, I have conjectured closed-form summations for two families of zeta series. Here in this work, I begin by using a known formula by Wilton to prove those conjectures. As example of applications, some special cases are explored, yielding rapidly converging series representations for the Ap\'{e}ry constant, $\zeta(3)$, and the Catalan constant, $G = \beta(2)$. Interestingly, our series for $\,\zeta(3)\,$ converges faster than that used by Ap\'{e}ry in his irrationality proof (1978). Also, our series for $\,G\,$ converges faster than a celebrated one discovered by Ramanujan (1915). At last, I present a simpler, more direct proof for a recent theorem by Katsurada which generalizes the above results.