On Evaluation of Zeta and Related Functions by Abstract Operators

TL;DR

This paper introduces a novel operator-based approach to evaluate Zeta and related functions, deriving new series expansions with rapid convergence across the complex plane and exploring their number theoretical properties.

Contribution

It develops an abstract operator framework to express Zeta and related functions, leading to new series expansions and insights into their number theoretical aspects.

Findings

Derived new rapidly converging series for ta(2n+1) and ta(2n+1)

Established series with order estimate O(m^{-2k}  k^{-2n+1})

Extended the evaluation of Zeta functions across the complex plane.

Abstract

Building on the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, 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: \[O(m^{-2k}\cdot k^{-2n+1})\qquad(k\rightarrow\infty;\quad m=3,4,6).\]