Parseval's Identity and Values of Zeta Function at Even Integers

TL;DR

This paper uses Parseval's identity applied to Bernoulli polynomials to derive explicit formulas for the Riemann zeta function at even positive integers, providing a novel approach to a classical problem.

Contribution

It introduces a new method leveraging Parseval's identity to compute zeta function values at even integers, expanding the toolkit for analytical evaluations.

Findings

Derived explicit formulas for zeta at even integers

Connected Bernoulli polynomials with zeta function values

Provided a new proof technique for classical results

Abstract

Historically known as the Basel problem, evaluating the Riemann zeta function at two has resulted in numerous proofs, many of which have been generalized to compute the function's values at even positive integers. We apply Parseval's identity to the Bernoulli polynomials to find such values.