From: Robin Chapman Subject: Re: Ramanujan Tau Function Date: Wed, 03 Nov 1999 08:30:04 GMT Newsgroups: sci.math To: ap512@columbia.edu In article , Amanda Pack wrote: > i must have miscopied an equation regarding the tau function. i have > written in my notes that > > tau(p) x tau(p^k) = tau(p^(k+1)) + p x tau(p^(k-1)) p=prime > > but the formula does not work. i've looked it up in a couple of books and > i haven't found it yet. does anyone know what the right equation may be? See Serre's A Course in Arithmetic. The formula is tau(p^{n+1}) = tau(p) tau(p^n) - p^11 tau(p^{n-1}). (*) One can remember this from the fact that the Dirichlet series sum_{n=1}^infinity tau(n)/n^s has the Euler product product_p (1 - tau(p)/p^s + p^11/p^{2s})^{-1}. The multiplicativity of tau and (*) follow from the fact that up to scalar factors, the modular form Delta is the only cusp form of weight 12 for the whole modular group SL(2,Z) and so is an eigenform for all the Hecke operators. -- Robin Chapman http://www.maths.ex.ac.uk/~rjc/rjc.html "`Well, I'd already done a PhD in X-Files Theory at UCLA, ...'" Greg Egan, _Teranesia_ Sent via Deja.com http://www.deja.com/ Before you buy.