Some series representing the eta function for $\Re s>0$

TL;DR

This paper presents new series representations for the Dirichlet eta function and related functions in the half-plane  s>0, using geometric series with controllable convergence properties.

Contribution

It introduces series representations dominated by geometric series for the eta function and related functions, with analysis of convergence and computational complexity.

Findings

Series representations have arbitrarily small convergence ratios.

Cost for each new term initially appears linear but may grow quadratically.

Number of terms needed increases linearly with the imaginary part of s.

Abstract

We represent the Euler alternating series (sometimes called the "Dirichlet eta function"), and generally $(b^s-b)\zeta(s)/b^s$ for $b>1$ an integer, in the half-plane $\Re s>0$, via series dominated by geometric series, with arbitrarily small convergence ratio (up to the prize of a longer first approximation). Due to the underlying recurrence, the cost for each new term is at first sight linearly increasing, so the cost appears to be quadratic in the number of terms kept. And the number of terms needed to achieve a given target precision increases linearly with the imaginary part of $s$.