Generalized rational zeta series for $\zeta(2n)$ and $\zeta(2n+1)$

TL;DR

This paper develops new rational zeta series representations involving even and odd zeta values, introduces generalized series using Clausen functions, and proves a 2012 conjecture, expanding the understanding of zeta function relations.

Contribution

It introduces two new families of generalized rational zeta series involving $eta$ and $ heta$ functions, and proves a conjecture by Lima from 2012.

Findings

Derived rational zeta series with $eta(2k)$ and $ heta(2k+1)$ functions.

Established a second family of generalized rational zeta series using the Clausen function.

Proved Lima's 2012 conjecture as a special case of the developed series.

Abstract

In this paper, we find rational zeta series with $\zeta(2n)$ in terms of $\zeta(2k+1)$ and $\beta(2k)$, the Dirichlet beta function. We then develop a certain family of generalized rational zeta series using the generalized Clausen function and use those results to discover a second family of generalized rational zeta series. As a special case of our results from Theorem 3.1, we prove a conjecture given in 2012 by F.M.S. Lima. Later, we use the same analysis but for the digamma function $\psi(x)$ and negapolygammas $\psi^{(-m)}(x)$. With these, we extract the same two families of generalized rational zeta series with $\zeta(2n+1)$ on the numerator rather than $\zeta(2n)$.