Evaluating $\zeta(s)$ At Odd Positive Integers Using Automatic Dirichlet Series

TL;DR

This paper constructs a Dirichlet series using Thue-Morse and paperfolding sequences to evaluate the Riemann zeta function at odd positive integers, offering an alternative proof of a 2015 result.

Contribution

It introduces a novel Dirichlet series approach to evaluate zeta at odd integers and provides an alternative proof of a known result.

Findings

Dirichlet series expressed as a linear combination of zeta at odd integers

New proof of a 2015 result by Allouche and Sondow

Connections between automatic sequences and zeta function evaluations

Abstract

In this paper, we use the Thue-Morse sequence and the paperfolding sequence to build a Dirichlet series that evaluates to a linear combination of the Riemann zeta function at odd positive integers and odd powers of $\pi$. In doing so, we also provide an alternative proof of a 2015 result by Allouche and Sondow.