A Comment on the Sums $\sum_{n \in \mathbb{Z}} \frac{(-1)^{nk}}{(an+1)^k}$

Abstract

We recall a proof of Euler's identity $\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$ involving the evaluation of a double integral. We extend the method to find Hurwitz Zeta series of the form $S(k,a)=\sum_{n \in \mathbb{Z}} \frac{(-1)^{nk}}{(an+1)^k},$ where $a \in \mathbb{N} \setminus \lbrace 1 \rbrace$ and $k \in \mathbb{N}.$ In particular, we consider a general $k$-dimensional integral over $(0,1)^k$ that equals the series representation $S(k,a).$ Then we use an algebraic change of variables that diffeomorphically maps $(0,1)^k$ to a $k$-dimensional hyperbolic polytope. We interpret the integral as a sum of two probabilities, and find explicit representations of such probabilities with combinatorial techniques.