A closed form expression for Zeta(2) and other even integer inputs of the Riemann Zeta function was found by Euler in the 1730’s. This article is going to use Mathematica to re-create the solution that Euler derived. The derivation below is based on the article in Wikipedia on the Basel Problem. Mathematica helped me see the pattern to solve Zeta(4) and in principle any positive even value. I don’t know if this is the approach Euler used to solve for all even integers. The Wikipedia article did not get into how Euler solved for the larger even integers.

The function I found in Mathematica which expands the infinite product (for a finite subset of the infinite product) is the Expand[] function. Unfortunately it also collects the terms for like powers of x. So one cannot see the first few terms of the zeta series evident in the product transformed into a series. So I had to get a little creative on how to show those terms. Read more…