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Distribution of Coefficients of Modular Forms and the Partition Function (1102.4400v2)
Published 22 Feb 2011 in math.NT
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 $1\le r\le\ellj$, $$\sharp{1\le n\le X\ |\ p(n)\equiv r\pmod{\ellj}}\gg_{s,r,\ellj}\frac{\sqrt X}{\log X}(\log\log X)s.$$