Geodesic
On the Sizes of Gaps in the Fourier Expansion of Modular Forms
Canadian journal of mathematics, Tome 57 (2005) no. 3, pp. 449-470

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Let $f\,=\,\sum{_{n=1}^{\infty }{{a}_{f}}\left( n \right){{q}^{n}}}$ be a cusp form with integer weight $k\,\ge \,2$ that is not a linear combination of forms with complex multiplication. For $n\,\ge \,1$ , let $${{i}_{f}}\left( n \right)\,=\,\left\{ _{0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise}\text{.}}^{\max \left\{ i\,:\,{{a}_{f}}\left( n+j \right)\,=\,0\,\text{for}\,\text{all}\,0\,\,\le \,j\,\le \,i \right\}\,\,\,\,\,\,\,\text{if}\,{{\text{a}}_{f}}\left( n \right)\,=\,0,} \right.\,$$ Concerning bounded values of ${{i}_{f}}\left( n \right)$ we prove that for $\in \,>\,0$ there exists $M\,=\,M\left( \in ,\,f \right)$ such that $\#\left\{ n\,\le \,x\,:\,{{i}_{f}}\left( n \right)\,\le \,M \right\}\,\ge \,\left( 1-\in\right)x.$ Using results of Wu, we show that if $f$ is a weight 2 cusp form for an elliptic curve without complex multiplication, then ${{i}_{f}}\left( n \right)\,{{\ll }_{f,\in }}\,{{n}^{\frac{51}{134}+\in }}$ . 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 ${{i}_{f}}\left( n \right)$ on the average using well known bounds on the Riemann Zeta function.
DOI : 10.4153/CJM-2005-019-7
Mots-clés : 11F30 
Alkan, Emre. On the Sizes of Gaps in the Fourier Expansion of Modular Forms. Canadian journal of mathematics, Tome 57 (2005) no. 3, pp. 449-470. doi: 10.4153/CJM-2005-019-7
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