All Questions
Tagged with analytic-number-theory modular-forms
19 questions
23
votes
3
answers
2k
views
Why are values of Eisenstein $E_2^*$ algebraic integers?
I'm looking for a proof that the following term is an algebraic integer whenever $\tau_N=\frac{N+\sqrt{-N}}{2}$ is a quadratic irrationality with class number $1$:
$$A_N:=\sqrt{-N}\cdot\frac{E_2(\...
- 598
6
votes
1
answer
600
views
How to compute Coefficients in Chudnovsky's Formula?
My aim is to understand all three coefficients arising in the Chudnovsky-Formula (see also Question 300385). Two of them are easily computed, but I failed with the third:
It is known that for all $\...
- 598
27
votes
4
answers
2k
views
Which quaternary quadratic form represents $n$ the greatest number of times?
Let $Q$ be a four-variable positive-definite quadratic form with integer coefficients and let $r_{Q}(n)$ be the number of representations of $n$ by $Q$. The theory of modular forms explains how $r_{Q}(...
- 20.4k
15
votes
1
answer
680
views
Is $\eta(\tau)^2$ a modular form of weight 1 on $\Gamma(12)$?
As we know, the Dedekind eta function $\eta(\tau)$ acquires a phase $\exp(2\pi i/24)$ under the modular transformation: $\tau \rightarrow \tau+1$. Therefore $\eta(\tau)^2$ is invariant under $\tau \...
- 153
12
votes
3
answers
2k
views
Kronecker's Jugendtraum for real quadratic fields?
Kronecker's Jugendtraum (or Hilbert's 12'th problem) is to find abelian extensions of arbitrary number fields by adjoining `special' values of transcendental functions. The Kronecker-Weber theorem was ...
- 1,905
38
votes
4
answers
6k
views
Modular forms and the Riemann Hypothesis
Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions?
What I'm thinking of is this: under the Mellin transform, the Riemann zeta function $...
- 889
21
votes
1
answer
1k
views
Why does this quasi-modular function have integral values?
It is a well-known result that the modular function $1728J(\tau) := \frac{1728E_4(\tau)^3}{E_4(\tau)^3-E_6(\tau)^2}$ has integral values if $\tau$ has class number 1 - for example at $\tau_{163}:=\...
- 598
15
votes
3
answers
1k
views
There is no lattice in PSL(2,R) which contains PSL(2,Z) properly?
How can I see that there is no lattice in $G=\mathrm{PSL}_2( \mathbb{R})$ which contains $\Gamma_1=\mathrm{PSL}_2( \mathbb{Z})$ properly, or equivalently, that $X_1 =\mathrm{PSL}_2(\mathbb{Z}) \...
- 11.2k
14
votes
4
answers
3k
views
Jacobi's theorem on sums of two squares (reference request)
One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals
$$4(d_1(n)-d_3(n)),$$
where the function $d_i$ counts the number ...
- 3,062
10
votes
2
answers
705
views
Averages over integer points of the sphere
A paper of William Duke proves that integer points on the sphere are equidistributed:
$$ V_n = \{ (x,y,z) \in \mathbb{Z}^2 : x^2 + y^2 + z^2 = n \}. $$
Up to reflections across the $x$, $y$ and $z$ ...
- 22.8k
9
votes
2
answers
556
views
Holomorphic Hoffstein-Lockhart
In the article Hoffstein, Jeffrey; Lockhart, Paul "Coefficients of Maass forms and the Siegel zero." Ann. of Math. (2) 140 (1994), no. 1, 161–181, it is stablished a good bound for the Petersson norm ...
- 243
7
votes
1
answer
811
views
Alternative way to prove the functional equation for Eisenstein series?
Let $E(z,s):=\pi^{-s}\Gamma (s) \sum_{(m,n)=1}\frac{y^s}{|mz+n|^{2s}}$ be the real-analytic Eisenstein series.
It satisfies the functional equation $E(z,s)=E(z,1-s)$ with two poles at $s=0,1$.
The ...
- 3,804
6
votes
1
answer
943
views
Voronoi formula for the symmetric $L$-function with level $N $
Sorry to disturb. Does any experts here know something upon the Voronoi type for the symmetric $L$-functions$$\sum_{n\le X} A_F(1,n)e\left ( \frac{an}{c}\right)=?$$
Here $F$ is a symmetric-lift of a $...
- 353
6
votes
1
answer
2k
views
The connection between the Weil conjectures and Ramanujan's conjecture
I'm writing an essay about Ramanujan's conjecture and have some questions:
1 How is Ramanujan's conjecture connected with the Weil conjectures?
2 How could Ramanujan's conjecture be assumed true or ...
user108974
6
votes
1
answer
305
views
Are the Fourier coefficients of $\eta(q^m)^m / \eta(q)$ non-negative?
In this paper, the following result is proved.
For any prime $p$, all the Fourier coefficients of
$$\eta(q^p)^p / \eta(q) = q^{\frac{p^2-1}{12}} \prod_{n=1}^\infty (1 - q^{pn})^p (1 - q^{n})^{-1}$$
...
- 427
5
votes
2
answers
433
views
Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms
Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients.
Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know the ...
- 153
4
votes
2
answers
1k
views
Real character modular forms: Fourier coefficient real?
Let $f$ be a modular form of level $N$ and real character $\chi$ of mod $N$ and weight $k$.
Does the Fourier coefficient or hecke-eigenvalue of $f$ have to be real?
What I knew is that if $N=1$ and $\...
- 3,804
3
votes
3
answers
483
views
Growth of the coefficients of the inversion of the $j$-invariant function
We have the $j$-invariant defined as
I have that
$$
j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k,
$$
where $q=e^{-2\pi t}$ ($\tau=it$) and $c_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$.
The inversion ...
- 515
1
vote
0
answers
200
views
Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$
I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is striclty greater than 1(...
- 11