Notes On An Approach To Apery's Constant

TL;DR

This paper explores geometric approaches to understanding Apery's constant and extends the geometric interpretations of the Riemann zeta function to higher arguments, building on prior visual explanations of the Basel problem.

Contribution

It introduces a geometric perspective to analyze Apery's constant and attempts to generalize visual interpretations of the zeta function beyond the Basel problem.

Findings

Geometric interpretation of $eta(2)$ related to pi squared

Extension of geometric methods to $eta(3)$ and higher zeta values

Discussion of limitations in visualizing $eta(n)$ for n > 3

Abstract

The Basel problem, solved by Leonhard Euler in 1734, asks to resolve $\zeta(2)$, the sum of the reciprocals of the squares of the natural numbers, i.e. the sum of the infinite series: \begin{equation} \sum_{i=1}^{\infty}\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots\notag \end{equation} The same question is posed regarding the summation of the reciprocals of the cubes of the natural numbers, $\zeta(3)$. The resulting constant is known as Apery's constant. A YouTube channel, 3BlueBrown, produced a video entitled, "Why is pi here? And why is it squared? A geometric answer to the Basel problem". The video presents the work of John W\"astlund. The equations can be extended to $\zeta(n)$, but the geometric argument is lost. We try to explore these equations for $\zeta(n)$.