Evaluating $\hat{I}\P (s)$ At Odd Positive Integers Using Automatic Dirichlet Series

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 $\ddot{I}€$. In doing so, we also provide an alternative proof of a 2015 result by Allouche and Sondow.