From: xxyzz
Newsgroups: sci.math,sci.math.symbolic
Subject: Re: clever double integral for zeta(2)
Date: Fri, 13 Nov 1998 06:05:30 -0500
Robin Chapman wrote:
> In article ,
> fiedorow@math.ohio-state.edu (Zbigniew Fiedorowicz) wrote:
> > In article <364B0272.220E59D9@nhh.no>, Per Erik Manne
> wrote:
> >
> > > I believe the other proof of the same kind was based on the
> > > integral of 1/(1-xy) over the same unit square. Here the correct
> > > substitution was u = x+y and v = x-y, and the procedure otherwise
> > > not too dissimilar.
> > > --
> >
> > Thanks, this was the proof I had in mind.
> >
> > Does anyone recall the famous mathematician who supposedly came up
> > with this idea?
>
> IIRC this proof appeared in a note by Tom Apostol in the Mathematical
> Intelligencer c. 1983. I don't know whether the proof was original to him.
>
I may not know how to spell "Riemann" :) but I do know that F. Beukers has a paper
that was published in the Bulletin of the London Math Society, 11 (1979),
268-272. He uses these integrals not only to get zeta(2) is irrational but he
also gets zeta(3) being irrational by considering an integral of the type
(log(xy))/(1-xy). The paper IMHO is a "must read" for people interested in
irrational and transcendental numbers.