The number of nonzero coefficients of modular forms (modp)

Bellaïche, Joël; Soundararajan, Kannan

doi:10.2140/ant.2015.9.1825

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The number of nonzero coefficients of modular forms $(\mathrm{mod} p)$

Joël Bellaïche and Kannan Soundararajan

Vol. 9 (2015), No. 8, 1825–1856

DOI: 10.2140/ant.2015.9.1825

Abstract

Let $f = \sum_{n = 0}^{\infty} a_{n} q^{n}$ be a modular form modulo a prime $p$ , and let $π (f, x)$ be the number of nonzero coefficients $a_{n}$ for $n < x$ . We give an asymptotic formula for $π (f, x)$ ; namely, if $f$ is not constant, then

π (f, x) \sim c (f) \frac{x}{{(log x)}^{α (f)}} {(log log x)}^{h (f)},

where $α (f)$ is a rational number such that $0 < α (f) \leq 3 ∕ 4$ , $h (f)$ is a nonnegative integer and $c (f)$ is a positive real number. We also discuss the equidistribution of the nonzero values of the coefficients $a_{n}$ .

Keywords

modular forms modulo $p$, Hecke operators, Selberg–Delange's method

Mathematical Subject Classification 2010

Primary: 11F33

Secondary: 11F25, 11N25, 11N37

Milestones

Received: 29 October 2014

Revised: 5 July 2015

Accepted: 3 August 2015

Published: 29 October 2015

Authors

Joël Bellaïche
Department of Mathematics Brandeis University 415 South Street Waltham, MA 02453 United States

Kannan Soundararajan
Department of Mathematics Stanford University 450 Serra Mall, Building 380 Stanford, CA 94305-2125 United States

Vol. 9, No. 8, 2015