Distribution of powers of the partition function modulo \(\ell ^j\) (Q838461)

A generalization of M. Newman's conjecture for the partition function \(p(n)\) states that if \(M\) and \(r\) are given positive integers, then for every integer \(s\) there are infinitely many nonnegative integers \(n\) such that \(p_r(n)\equiv s\pmod M\), where \(p_r(n)\) counts the number of \(r\)-colored partitions of \(n\). The authors prove this conjecture when \(M= p^j\), where \(p\) is prime and the pairs \((r,p)\) are subject to certain restrictions.