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 $a\in \mathbb{N}\setminus \{1\}$ 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.

References in corpus (2)

• Evaluation of Harmonic Sums with Integrals
• Comment on the sums $S(n)=\sum _{k=-\infty }^{\infty }\frac{1}{(4k+1{)}^n}$