New Proofs of the Basel Problem using Stochastic Processes

TL;DR

This paper explores new proofs of the Basel problem by leveraging stochastic processes, providing historical context and novel mathematical approaches to derive the sum of the reciprocals of squares.

Contribution

It introduces stochastic process-based methods to generate new proofs of the Basel problem, expanding the mathematical tools used for this classic problem.

Findings

New proofs of the Basel problem using stochastic processes

Historical perspective on the problem's significance

Potential for generating further mathematical proofs

Abstract

The number $\frac{\pi ^{2}}{6}$ is involved in the variance of several distributions in statistics. At the same time it holds $\sum\nolimits_{k=1}^{\infty }k^{-2}= \frac{\pi ^{2}}{6}$, which solves the famous Basel problem. We first provide a historical perspective on the Basel problem, and second show how to generate further proofs building on stochastic processes.