On the Multiple Zeta Values $\zeta(\{2\}^k)$

TL;DR

This paper evaluates specific multiple zeta values using combinatorial and elementary series techniques, revealing new constants related to pi and connecting trigonometric properties with infinite product representations.

Contribution

It introduces a novel proof of the evaluation of multiple zeta values $ ext{zeta}( ext{} extbraceleft 2 extbraceright^k)$ through combinatorial bijections and series, and defines new constants linked to pi.

Findings

Evaluation of $ ext{zeta}( extbraceleft 2 extbraceright^k)$ values

Introduction of pi-frequency and pi-amplitude constants

Connection between these constants and the geometric definition of pi

Abstract

We evaluate the multiple zeta values $\zeta(\{2\}^k)$ by proving a certain factorization property. The proof uses a combinatorial bijection and elementary telescoping series. We show how the infinite product for the sine function in fact implies its power series and other trigonometric properties. We define two constants, which we call pi-frequency and pi-amplitude, and show that they are equal and satisfy the geometric definition of pi arising from the circumference of the circle.