Distribution of Coefficients of Modular Forms and the Partition Function

Details Distribution of Coefficients of Modular Forms and the Partition Function Distribution of Coefficients of Modular Forms and the Partition Function

2011-02-22

arxiv.org/abs/1102.4400v2

Mathematics

Number Theory

8pages

Scientific paper

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 $1\le r\le\ell^j$, $$\sharp\{1\le n\le X\ |\ p(n)\equiv
r\pmod{\ell^j}\}\gg_{s,r,\ell^j}\frac{\sqrt X}{\log X}(\log\log X)^s.$$

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