Rational zeta series for zeta(2n) and zeta(2n+1)

Abstract:

Abstract: I will begin by using the cotangent function to find rational zeta series with $\zeta(2n)$ in terms of $\zeta(2k+1)$ and $\beta(2k)$, the Dirichlet beta function. I then develop a certain family of generalized rational zeta series using the generalized Clausen function $\Clausen_{m}(x)$ and use those results to discover a second family of generalized rational zeta series. As a special case of my results from Theorem 3.1, I prove a conjecture given in 2012 by F.M.S. Lima. Later, I use the same analysis but for the digamma function $\psi(x)$ and negapolygammas $\psi^{(-m)}(x)$. With these, I extract the same two families of generalized rational zeta series with $\zeta(2n+1)$ on the numerator rather than $\zeta(2n)$. Afterwards, I look into the applications of these rational zeta series and how they are related to other special functions such as the multiple zeta function.