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.