Special values of the Riemann zeta function via arcsine random variables

Abstract

In this paper, using arcsine variables, we get a new elementary proof of $\zeta \left(2\right)={\pi }^2/6$, known as the Basel problem, and the Euler formula. Using exponential variables, we get an Euler-like formula. We can also solve the Basel problem by using Wigner’s semicircle law and the Legendre generating function.