From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: zeta
Date: 12 Jul 1998 05:20:01 GMT

Matthew Sullivan wrote:
> A couple questions about zeta, first off how did Euler prove that the
> summation from 1 to infinity of 1/n^2 = p^2/6?  

Peter Ammon  <Peter_Ammon@rocketmail.com> wrote:
>It can be done using Fourier series (kind of like a power series except
>is uses sines and cosines).  I've heard rumors of an easy proof, but
>never seen it. 

I am told Euler suggested this:

If  f = 1 + a1*x + a2*x^2 + ...  is a polynomial with nonzero roots, then
the sum of the reciprocals of the roots is  -a1, the sum of their
squares is  a1^2-2 a2, and so on (the usual relations between symmetric
functions and powers).

Apply this to  f(x) = sin(x)/x, which vanishes precisely at  x = n * Pi,
n = +-1, +-2, ..., and which may be expressed as  1 - (1/3) x^2 + ...

This 'proof' must really be right, since it gives the right answers
for the other values of  zeta(2k), too!

dave