On Evaluation of Zeta and Related Functions by Abstract Operators

Building on the mapping relations between analytic functions and periodic functions using the abstract operators $\cos (h{\mathrm{\partial }}_x)$ and $\sin (h{\mathrm{\partial }}_x)$, and by defining the Zeta and related functions including the Hurwitz Zeta function and the Dirichlet L-function in the form of abstract operators, we have obtained many new series expansions associated with these functions on the whole complex plane, and investigate the number theoretical properties of them, including some new rapidly converging series for $\eta (2n+1)$ and $\zeta (2n+1)$. For $n\in \mathbb{N}$, each of these series representing $\zeta (2n+1)$ converges remarkably rapidly with its general term having the order estimate: $$O(m^{-2k}\cdot k^{-2n+1})(k\to \mathrm{\infty };m=3,4,6)\mathrm{.}$$