The computation of $ζ(2k)$, $β(2k+1)$ and beyond by using telescoping series

TL;DR

This work presents an accessible, elementary framework for evaluating classical zeta and Dirichlet beta values through telescoping-series proofs that express $\zeta(2k)$ and $\beta(2k+1)$ in terms of Bernoulli and Euler polynomials. It develops auxiliary integrals and leverages generating-function structures to derive closed forms, notably $\zeta(2k)=\frac{(-1)^{k-1}2^{2k-1}\pi^{2k}}{(2k)!}B_{2k}$ and $\beta(2k+1)=\frac{(-1)^k\pi^{2k+1}}{2^{2k+2}(2k)!}E_{2k}$, while also obtaining integral representations for $\zeta(2k+1)$ and $\beta(2k)$. The paper then generalizes these telescoping ideas to Apostol-Bernoulli and Apostol-Euler polynomials, enabling exact sums over $\mathbb{Z}$ and alternating/positive series via explicit formulas such as $\mathcal{Z}(k;\mu)=\frac{1}{2\cdot k!} i^k e^{i\mu/2}\mathcal{E}_k(1/2;e^{i\mu})$ and its real variants, along with associated partial-fraction identities. Overall, the approach provides self-contained, elementary proofs and broad generalizations for a range of zeta-type sums that illuminate the connections between special polynomials and classical number-theoretic sums.

Abstract

We present some simple proofs of the well-known expressions for \[ ζ(2k) = \sum_{m=1}^\infty \frac{1}{m^{2k}}, \qquad β(2k+1) = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^{2k+1}}, \] where $k = 1,2,3,\dots$, in terms of the Bernoulli and Euler polynomials. The computation is done using only the defining properties of these polynomials and employing telescoping series. The same method also yields integral formulas for $ζ(2k+1)$ and $β(2k)$. In addition, the method also applies to series of type \[ \sum_{m\in\mathbb{Z}} \frac{1}{(2m-μ)^s}, \qquad \sum_{m\in\mathbb{Z}} \frac{(-1)^m}{(2m+1-μ)^s}, \] in this case using Apostol-Bernoulli and Apostol-Euler polynomials.

Theorems & Definitions (30)

• Remark 1.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3: Euler, 1740
• proof
• Remark 3.4
• Lemma 4.1
• Remark 4.2