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Archiv der Mathematik

, Volume 98, Issue 4, pp 307–315 | Cite as

Distribution of the coefficients of modular forms and the partition function

  • Shi-Chao Chen
Article

Abstract

Let 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 j and prove that for each integer 1 ≤ r <  j ,
$$\sharp\{1\le n\le X\ |\ p(n)\equiv r\pmod{\ell^j} \}\gg_{s,r,\ell^j} \frac{\sqrt X}{\log X}(\log\log X)^s.$$

Mathematics Subject Classification

Primary 11P83 Secondary 11F33 

Keywords

Modular forms The ordinary partition function 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Information Sciences, Institute of Contemporary MathematicsHenan UniversityKaifengPeople’s Republic of China

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