Newman's conjecture for the partition function modulo integers with at least two distinct prime divisors

Abstract

Let M be a positive integer and $p\left(n\right)$ be the number of partitions of a positive integer n. Newman's Conjecture asserts that for each integer r, there are infinitely many positive integers n such that$p\left(n\right)\equiv r\left(\mathrm{mod}M\right).$ For a positive integer d, let $B_d$ be the set of positive integers M such that the number of prime divisors of M is d. In this paper, we prove that for each positive integer d, the density of the set of positive integers M for which Newman's Conjecture holds in $B_d$ is 1. Furthermore, we study an analogue of Newman's Conjecture for weakly holomorphic modular forms on ${\Gamma }_0\left(N\right)$ with nebentypus, and this applies to t-core partitions and generalized Frobenius partitions with h-colors.