On the Sizes of Gaps in the Fourier Expansion of Modular Forms

http://dx.doi.org/10.4153/CJM-2005-019-7
Canad. J. Math. 57(2005), 449-470

  Published:2005-06-01
 Printed: Jun 2005

Let $f= \sum_{n=1}^{\infty} a_f(n)q^n$ be a cusp form with integer weight $k \geq 2$ that is not a linear combination of forms with complex multiplication. For $n \geq 1$, let $$ i_f(n)=\begin{cases}\max\{ i : a_f(n+j)=0 \text{ for all } 0 \leq j \leq i\}&\text{if $a_f(n)=0$,}\\ 0&\text{otherwise}.\end{cases} $$ Concerning bounded values of $i_f(n)$ we prove that for $\epsilon >0$ there exists $M = M(\epsilon,f)$ such that $\# \{n \leq x : i_f(n) \leq M\} \geq (1 - \epsilon) x$. Using results of Wu, we show that if $f$ is a weight 2 cusp form for an elliptic curve without complex multiplication, then $i_f(n) \ll_{f, \epsilon} n^{\frac{51}{134} + \epsilon}$. Using a result of David and Pappalardi, we improve the exponent to $\frac{1}{3}$ for almost all newforms associated to elliptic curves without complex multiplication. Inspired by a classical paper of Selberg, we also investigate $i_f(n)$ on the average using well known bounds on the Riemann Zeta function.