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


Published:20050601
Printed: Jun 2005
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Abstract
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.