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

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.