Some Rapidly Converging Series for $\zeta(2n+1)$ from Abstract Operators

TL;DR

The paper introduces rapidly converging series representations for the Riemann Zeta function at odd integers using abstract operators, providing a new method for summing Dirichlet series and extending results across the complex plane.

Contribution

It presents a novel approach using abstract operators to derive series for ta(2n+1) with rapid convergence, generalizing to the entire complex plane.

Findings

Series for ta(2n+1) converge quickly with explicit error estimates.

Method based on mapping between analytic and periodic functions using abstract operators.

Generalized results applicable to the whole complex plane.

Abstract

The author derives new family of series representations for the values of the Riemann Zeta function $\zeta(s)$ at positive odd integers. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with its general term having the order estimate: $$O(m^{-2k}\cdot k^{-2n+1})\qquad(k\rightarrow\infty;\quad m=3,4,6).$$ The method is based on the mapping relationships between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, including the mapping relationships between power series and trigonometric series, if each coefficient of a power series is respectively equal to that of a trigonometric series. Thus we obtain a general method to find the sum of the Dirichlet series of integer variables. By defining the Zeta function in an abstract operators form, we have further generalized these results on the whole complex plane.