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

[摘要] Let $f= sum_{n=1}^{infty} a_f(n)q^n$ be a cusp form with integerweight $k geq 2$ that is not a linear combination of forms withcomplex multiplication. For $n geq 1$, let $$ i_f(n)=egin{cases}max{ i : a_f(n+j)=0 ext{ for all } 0 leq j leqi}&ext{if $a_f(n)=0$,}\0&ext{otherwise}.end{cases} $$ Concerning bounded valuesof $i_f(n)$ 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 2cusp form for an elliptic curve without complex multiplication, then$i_f(n) ll_{f, epsilon} n^{frac{51}{134} + epsilon}$. Using aresult of David and Pappalardi, we improve the exponent to$frac{1}{3}$ for almost all newforms associated to elliptic curveswithout complex multiplication. Inspired by a classical paper ofSelberg, we also investigate $i_f(n)$ on the average using well knownbounds on the Riemann Zeta function.

[学科分类] 数学（综合）