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: $$X\ll \dfrac{\left(\sum_{n\le X}a(n)\right)^2}{\underset{n\le X}{\sum} a(n)^2}$$
Thanks !
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1$\begingroup$ I agree with asd's answer that the ratio is surely $o(X)$ in many cases (e.g. for Hecke cusp forms), although I have not seen this in print and it might be somewhat tiresome to derive it from scratch. At any rate, the recent paper of Hulse-Kiral-Kuan-Lim (Int. J. Number Theory 8 (2012), 749-762) certainly suggests that the ratio is $O(X^c)$ for any $c>1/2$. See especially (3.3) and (3.10) in their paper, but note that $t$ is restricted to square-free integers in these relations, while $a(t)$ stands for a normalized Fourier-coefficient. $\endgroup$– GH from MOCommented Aug 3, 2016 at 18:06
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$\begingroup$ @GHfromMO, Thank you very much for the reference. $\endgroup$– user95750Commented Aug 3, 2016 at 18:11
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3$\begingroup$ Some other thoughts: for integral weight forms, there are well-known techniques to upper bound the sum $\sum_{n\leq X}a(n)$ and to evaluate asymptotically the sum $\sum_{n\leq X}a(n)^2$ (at least when $\chi$ is a quadratic character which forces $a(n)$ to be real). I am sure these techniques can be extended to half-integral weight forms. Regarding the first sum (for integral weight forms), see Section 2.7 in Harcos-Michel: The subconvexity problem for Rankin-Selberg L-functions II. The necessary upper bounds for $f(x+iy)$ can probably be generalized from arxiv.org/abs/1504.08246 $\endgroup$– GH from MOCommented Aug 3, 2016 at 20:48
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$\begingroup$ @GHfromMO, That's very helpful thank you very much. $\endgroup$– user95750Commented Aug 3, 2016 at 21:16
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If your $f$ is a Hecke-cusp form then the estimate is certainly false, and the ratio is $o(X)$, since the coefficients $a(n)$ oscillate and there is a lot of cancellations in $\sum a(n)$. If $f$ is something like an Eisenstein series then it's probably true, because the coefficients are positive and mildly behaved. You should specify if you're interested in Hecke cusp form or Eisenstein series or if this is a general query.
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$\begingroup$ Thanks for your answer, I am interested to the general case $f\in M_{k+1/2}(\Gamma_0(4N),\chi)$ $\endgroup$ Commented Aug 3, 2016 at 16:43
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2$\begingroup$ 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. $\endgroup$– asdCommented Aug 3, 2016 at 19:12
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$\begingroup$ 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. $\endgroup$ Commented Aug 3, 2016 at 19:53
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2$\begingroup$ @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. $\endgroup$ Commented Aug 3, 2016 at 20:52
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1$\begingroup$ @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. $\endgroup$– asdCommented Aug 3, 2016 at 23:42