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

TL;DR

The paper develops a convergent, geometry-dominated series representation for the Dirichlet eta function and its generalizations $\eta_b(s)$ in the half-plane $\Re s>0$, by splitting the Dirichlet sum into a finite initial block and a tail weighted by coefficients $c_{m,b}^*(s)$ defined through a linear recurrence. The main technique relies on a recursive definition of $c_{m,b}^*(s)$ and a tight analysis of their bounds, yielding explicit, $\sigma$-dependent controls and showing equality cases on vertical lines $s=\sigma+\frac{2\pi i k}{\log b}$. The proofs connect these coefficients to known structures from Burnolzeta and Bernoulli numbers, establishing absolute/uniform convergence of the expansion and reproducing $\eta(s)$ at $b=2$ while generalizing to $\eta_b(s)$ for any integer $b>1$. The results illuminate a practical, controllable path for evaluating Dirichlet eta values with arbitrary precision in $\Re s>0$, and they clarify the cost behavior with respect to the imaginary part of $s$. Overall, the work provides a rigorous, recurrence-based framework for eta-function computation with explicit bounds and convergence properties.

Abstract

We represent the Dirichlet eta function, and generally $(b^s-b)ζ(s)/b^s$ 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 $s$.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Proposition 1
• Remark 1
• Remark 2
• Proposition 2
• proof : Proof of Proposition \ref{['prop:1']}
• proof : Proof of Proposition \ref{['prop:2']}