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

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.