Timeline for Estimate the ratio $\dfrac{\left(\sum_{n\le X}a(n)\right)^2}{\underset{n\le X}{\sum} a(n)^2}$
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 8, 2016 at 22:19 | comment | added | reuns | @A.M.Amine for holomorphic Eiseinstein series $a(n) = \sigma_k(n)$, it is as easy as proving $h(\tau) = \sum_{n=-\infty}^\infty \frac{1}{n + \tau m}$ is a trigonometric function, and expanding this trigonometric function as a series $\sum_k c_k e^{2i \pi m k \tau}, Im(\tau) > 0$ | |
| Aug 4, 2016 at 0:10 | comment | added | GH from MO | @asd: I agree with your disagreement. Thanks for pointing this out! | |
| Aug 3, 2016 at 23:42 | comment | added | asd | @GH from MO: I disagree a little bit about the generalization to half-integral weight forms; for example the coefficients since of some half-integral weight Eisenstein series are class numbers of imaginary quadratic fields, so they're not completely straightforward, although still easy on average, since that would amount to the moment of an L-function at the edge of the critical strip. | |
| Aug 3, 2016 at 20:52 | comment | added | GH from MO | @A.M.Amine: For basis Eisenstein series the $a(n)$'s are explicit sums over the divisors of $n$, hence their moments can also be evaluated explicitly. Expressions for $a(n)$ in integral weight can be found in many textbooks, e.g. in Chapter 7 of Miyake: Modular Forms. Generalizing to half-integral weight probably causes no serious obstacles. | |
| Aug 3, 2016 at 19:53 | comment | added | user95750 | Could you suggest to me some references in which I can find tools to prove that $$X\ll\dfrac{\left(\sum_{n\le X}a(n)\right)^2}{\underset{n\le X}{\sum} a(n)^2}$$ for Eisenstein series. Many thanks. | |
| Aug 3, 2016 at 19:12 | comment | added | asd | In the general case you can write $a(n)$ as a combination of coefficients of Eisenstein series and Hecke cusp forms. In the first moment only the Einsteinstein series will survive, while in the second moment you'll have the squares of Eisenstein series and Hecke cusp forms that survive. So roughly speaking your ratio will be $\gg X$ if $f$ expressed as a combination of Eisenstein series and Hecke cusp forms, has a non-trivial contribution from Eisenstein series. There might be some pathologies to watch out, but heuristically that should be it. | |
| Aug 3, 2016 at 19:11 | vote | accept | user95750 | ||
| Aug 3, 2016 at 16:45 | review | First posts | |||
| Aug 3, 2016 at 16:56 | |||||
| Aug 3, 2016 at 16:43 | comment | added | user95750 | Thanks for your answer, I am interested to the general case $f\in M_{k+1/2}(\Gamma_0(4N),\chi)$ | |
| Aug 3, 2016 at 16:40 | history | answered | asd | CC BY-SA 3.0 |