Fourier Analysis and the closed form for the Zeta Function at even positive integers

TL;DR

This paper derives a simplified and concise closed-form expression for the Riemann zeta function at even positive integers using Fourier analysis and a summation identity for Fourier coefficients.

Contribution

It introduces a more straightforward method to obtain the closed form of the zeta function at even integers, improving upon existing proofs.

Findings

Closed form expression for ζ(2k) derived

Simpler and shorter proof compared to previous methods

Method based on Fourier coefficients and summation identities

Abstract

Using a summation identity obtained for the Fourier coefficients of $x^{2k}$, we derive a closed form expression for the zeta function at even positive integers, using a technique similar to one in an existing proof by Aladdi and Defant[1], but in a simpler and shorter way.