Tagged Questions
1
vote
1answer
36 views
Routine question about derivatives of automorphic forms being L^2
I consider Automorphic forms on $G = SL_2 (\mathbb{R})$, which are $\Gamma$-invariant, $K$-finite, $Z(g)$ finite, and of moderate growth. If I have such an automorphic form, which happens to be in ...
2
votes
1answer
64 views
Why does a certain equality in Iwaniec's Topics in classical automorphic forms hold?
I have been reading the book Topics in classical automorphic forms for a while, where I encountered a formula in the middle of Sec 8.6 (p143)
$$F(z;\beta,\gamma)=\frac ...
1
vote
1answer
95 views
Eisenstein series solution
Denote the function $$ \Psi (x,y)= y^{1/2+ik}+ \sum_{g\in SL(2,\Bbb Z)} \frac{y^{1/2+ik}} {|c_{g}z+d_{g}|^{1/2+ik}}\tag{1}$$
My question is if I can write the wave function in terms of the Eisenstein ...
5
votes
0answers
305 views
Residue of Rankin-Selberg L-function for non-trivial nebentypus
Let $f\in S_k(\Gamma_0(N),\chi)$ be a normalized holomorphic newform (i.e. weight $k$, level $N$, nebentypus $\chi$) and write its Fourier expansion as
$$ f(z)=\sum_{n\ge 1} ...
2
votes
1answer
89 views
Is $\eta^{24}(\tau)\,j(\tau) = {E_4}^3(q)$?
Given the j-function $j(\tau)$,
$j(\tau) = 1728J(\tau)$,
where $J(\tau)$ is Klein’s absolute invariant, the Dedekind eta function $\eta(\tau)$, and the following Eisenstein series,
$\begin{align}
...
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
$$ (1-n)\tau(n) = 24\sum_{j=1}^{n-1} \sigma(j)\tau(n-j).$$
...
8
votes
4answers
168 views
Representing a number as a sum of at most $k$ squares
Fix an integer $k >0 $ and would like to know the maximum number of different ways that a number $n$ can be expressed as a sum of $k$ squares, i.e. the number of integer solutions to
$$ n = x_1^2 + ...
8
votes
1answer
149 views
Converting an infinite product to sum; Ramanujan $\tau$ function
I've gotten what seems most of the way, but I'm quite stuck at this point.
Define $\tau(n)$ by
\begin{align*}
q\prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty\tau(n)q^n.
\end{align*}
...
1
vote
0answers
117 views
Bounds for Fourier coefficients of cusp forms
I've asked the background question here, which still left unanswered.
Now I have a more precise question. In my homework I've been asked to prove that
$$\left| \sum_{1\leq n \leq N} a_f (n)e^{2\pi i ...
5
votes
1answer
252 views
Cusp forms' Fourier coefficients sign changes
I need some clarification on the following, if possible: I have seen in that for every $ f \in S_k$ which Fourier transform is $\sum_{n=1}^\infty a(n)q^n$ there is an upper bound $\sum_{n=1}^N ...
