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, ...'"
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