Revisiting The Riemann Zeta Function at Positive Even Integers

TL;DR

This paper offers a new proof for the formula of the Riemann zeta function at positive even integers using Parseval's identity, enhancing understanding of its mathematical properties.

Contribution

It introduces a novel proof method for the zeta function at even integers based on Fourier analysis and Parseval's identity.

Findings

Confirmed the explicit formula for ζ(2k) using Fourier coefficients

Provided a new proof approach for a classical mathematical result

Enhanced understanding of the connection between Fourier analysis and special functions

Abstract

Using Parseval's identity for the Fourier coefficients of $x^k$, we provide a new proof that $\zeta(2k)=\dfrac{(-1)^{k+1}B_{2k}(2\pi)^{2k}}{2(2k)!}$.