All Questions
5 questions
3
votes
0
answers
235
views
Functional equation link two Dirichlet series
Let $f(z)=\sum_{n\ge 1}a(n)n^{(k-1/2)/2}e(nz)\in S_{k+1/2}(\Gamma_0(4N),\chi)$ be a cuspidal Hecke eigenform. Let
$$M(s)=\sum_{p\ge 2, \text{prime}}\frac{|a(p)|^2}{p^s}$$ and
$$R_f(s)=\sum_{n\ge 1}\...
3
votes
1
answer
344
views
Estimate the ratio $\dfrac{\left(\sum_{n\le X}a(n)\right)^2}{\underset{n\le X}{\sum} a(n)^2}$
Let $f=\sum_{n\ge 1} a(n)q^n\in M_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)$ be a modular form of half-integral wieght.
Can someone prove or disprove that:
$$X\ll \dfrac{\left(\sum_{n\le X}a(n)\right)^2}{\...
0
votes
1
answer
218
views
Clarification of the proof of the main theorem of the paper of Hulse et al
I am trying to understand some open steps in the following article The Sign of Fourier coefficients of Half-integral Weight Cusp Form by Hulse, Kiral, Kuan, and Lim, I find the following :
Let $f\...
8
votes
1
answer
739
views
Average of Fourier coefficients of a cusp form of half integral weight
Suppose $f$ is a cusp form of half integral weight $k$ w.r.t. the group $\Gamma_0(4)$ ($k$ is not very low, can assume $k \ge 11/2$), and $a_n$ is its Fourier coefficient. The Linnik bound says that ...
1
vote
1
answer
324
views
Off critical line zeros for half integer weight $L$-functions
Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put
$$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$
to be the $L$-function.
Further ...