Distribution of Coefficients of Modular Forms and the Partition Function

Abstract

Let $\ell \ge 5$ 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 $1\le r\le {\ell }^j$, $$♯\{1\le n\le X|p(n)\equiv r(\mathrm{mod}{\ell }^j)\}{\gg }_{s,r,{\ell }^j}\frac{\sqrt{X}}{\log X}(\log \log X{)}^s.$$

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