Distribution of the coefficients of modular forms and the partition function

摘要

Let ? ≥ 5 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.