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Distribution of the coefficients of modular forms and the partition function

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Abstract

Let ℓ 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 an odd positive integer M. As an application, we investigate the distribution of the ordinary partition function p(n) modulo ℓ j and prove that for each integer 1 ≤ r < ℓ 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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  1. Department of Mathematics and Information Sciences, Institute of Contemporary Mathematics, Henan University, Kaifeng, 475004, People’s Republic of China

    Shi-Chao Chen

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  1. Shi-Chao Chen
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Chen, SC. Distribution of the coefficients of modular forms and the partition function. Arch. Math. 98, 307–315 (2012). https://doi.org/10.1007/s00013-012-0375-1

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