Tagged Questions
7
votes
1answer
235 views
Ramanujan's Tau function, an arithmetic property
The problem:
Let $\tau(n)$ denote the Ramanujan $\tau$-function and $\sigma(n)$ be the sum of the positive divisors of $n$. Show that
$$ (1-n)\tau(n) = 24\sum_{j=1}^{n-1} \sigma(j)\tau(n-j).$$
...
9
votes
1answer
263 views
Relation between congruence subgroups. $\Gamma(M)\Gamma(N) = \Gamma(\gcd(M,N))$
I'm hoping there's a pleasant way to solve this one.
Prove that $\Gamma(M)\Gamma(N) = \Gamma(\gcd(M,N))$.
Showing that $\Gamma(M)\Gamma(N) \subset \Gamma(\gcd(M,N))$ is rather straight forward, ...
8
votes
1answer
149 views
Converting an infinite product to sum; Ramanujan $\tau$ function
I've gotten what seems most of the way, but I'm quite stuck at this point.
Define $\tau(n)$ by
\begin{align*}
q\prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty\tau(n)q^n.
\end{align*}
...
1
vote
0answers
117 views
Bounds for Fourier coefficients of cusp forms
I've asked the background question here, which still left unanswered.
Now I have a more precise question. In my homework I've been asked to prove that
$$\left| \sum_{1\leq n \leq N} a_f (n)e^{2\pi i ...
