Distribution of the coefficients of modular forms and the partition function

摘要

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