7
$\begingroup$

$E_a(z,s)$ denotes the Eisenstein series expanded at the cusp $a$. For each cusp $a=\frac{u}{w}$ of $\Gamma_0(N)$, we define the Eisenstein series $$ \begin{eqnarray}E_a(z,s)&=&\sum_{\tau\in\Gamma_a\backslash\Gamma}\Im(\sigma^{-1}_a\tau z)\\&=&\delta_{a\infty}y^s+\sqrt{\pi}\frac{\Gamma(s-\dfrac{1}{2})}{\Gamma(s)}\rho_a(s,0)y^{1-s}\\&+&\frac{2\pi^s\sqrt{y}}{\Gamma(s)}\sum_{m\neq0}|m|^{s-\frac{1}{2}}\rho_a(s,m)K_{s-\dfrac{1}{2}}(2\pi|m|y)e(mx), \end{eqnarray} $$ where $$\rho_a(s,m)=\left(\frac{(w,\frac{N}{w})}{wN}\right)^s\sum_{(\gamma,\frac{N}{w})=1}\frac{1}{\gamma^{2s}}\sum_{\delta(\bmod \gamma w),(\delta,\gamma w)=1, \delta \gamma\equiv u\left(\bmod(w,\dfrac{N}{w})\right)}e(-m\dfrac{\delta}{\gamma w}).$$

I would like to know what is the answer or relation between Riemann's $\zeta(s)$ of the following two L-series $\sum_{n>0}\dfrac{\rho_a(s,n)}{n^w}$ and $\sum_{n>0}\dfrac{\widetilde{\rho_a(s,n)}}{n^w}$ where $\widetilde{\rho_a(s,n)}=\dfrac{\rho_a(s,n)\zeta(2s)}{\sigma_{1-2s}(m)}$. Thanks.

  • 0
    +1 interesting. Where does it comes from and what have you tried so far?2012-06-26
  • 0
    Very good question +1.2012-06-26
  • 0
    @draks My questions come from the paper "multiple dirichlet series and shifted convolutions" written by Jeff Hoffstein. I have no idea about the answers.2012-06-26
  • 0
    On-line reference here: [Multiple Dirichlet series and Shifted Convolutions](http://arxiv.org/pdf/1110.4868v1.pdf) by Jeff Hoffstein.2012-06-26
  • 0
    @nglineline where in 104 pages are you? Page 12?2012-06-26
  • 1
    What does $\delta(\bmod \gamma w)$ under the inner sum mean?2012-06-26
  • 0
    @draks I got the information from page 12. δ(modγw) under the inner sum means the sum over the complete resdiue class of γw.2012-06-26
  • 0
    Have you tried taking the Mellin transform of the higher-order terms in the Fourier expansion (like calculating the $L$-function of a cusp form)? I'd start with $N=1$ case. If I get a chance, I'll add some details.2012-06-26
  • 0
    @BR For the case N=1, it is the classical Eisenstein series.2012-06-27
  • 0
    @nglineline, Yes, it is. The other cases will have a similar answer, but dealing with the different cusps can get complicated. If you think about this adelically, where there is only one cusp, you'll see that the general case is not much more difficult than the $N=1$ case.2012-06-27
  • 2
    @nglineline, Basically, for the $N=1$ case, you'll end up with something like $${\zeta(w+s)\zeta(w+1-s)\over\zeta(2s)}$$ For general $N$, $\zeta$ will be replaced by a Dirichlet $L$-function, and there will probably be some additional nuisance factors. Off the top of my head, I can't derive the correspondence between the Dirichlet character and the cusp (though I'm sure cusps in $\Gamma_0(N)$ correspond to Dirichlet characters of order $N$).2012-06-27

1 Answers 1