Some series representing the zeta function for $\Re s>1$

TL;DR

This paper introduces a series representation of the Riemann zeta function for real part greater than one, using geometrically decreasing bounds and recurrence relations, with limited practical use for large imaginary parts.

Contribution

It develops a new series representation of the zeta function based on recurrence relations and geometric bounds, extending previous tools for harmonic series evaluation.

Findings

Series converges for Re s > 1 with geometrically decreasing terms

Computational cost is at least quadratic in the number of terms

Method is practical for small imaginary parts, up to hundreds

Abstract

We represent the Riemann zeta function in the half-plane $\Re s >1$ via series whose terms admit geometrically decreasing bounds. Due to an underlying recurrence relation, which is used to compute coefficients entering into the terms, the cost is probably at least quadratic in the number of terms. And the number of terms needed to reach a given fixed-point precision grows linearly with the imaginary part, so, presumably, the usefulness is limited to small imaginary parts (up to the hundreds perhaps). The method is a development of tools introduced by the author for the evaluation of harmonic series with restricted digits in a given radix.