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 !



Estimate the ratio $\dfrac{\left(\sum_{n\le X}a(n)\right)^2}{\underset{n\le X}{\sum} a(n)^2}$
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. 


