Distribution of Coefficients of Modular Forms and the Partition Function

Abstract

Let $\ell\ge5$ be an odd prime and $j, s$ be positive integers. We study the distribution of the coefficients of integer and half-integral weight modular forms modulo odd positive integer $M$. As a consequence, we prove that for each integer $1\le r\le\ell^j$, $$\sharp\{1\le n\le X\ |\ p(n)\equiv r\pmod{\ell^j}\}\gg_{s,r,\ell^j}\frac{\sqrt X}{\log X}(\log\log X)^s.$$