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

TL;DR

This paper proves that Newman's conjecture holds for almost all integers with at least two prime factors and explores related conjectures for modular forms and special partition types.

Contribution

It establishes the density of integers satisfying Newman's conjecture within sets defined by prime divisor count and extends the conjecture to modular forms and specialized partitions.

Findings

Density of integers satisfying the conjecture is 1 in sets with fixed prime divisor count.

The conjecture holds for integers with at least two prime factors.

Extensions to modular forms and special partition classes are provided.

Abstract

Let $M$ be a positive integer and $p(n)$ 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 \[ p(n)\equiv r \pmod{M}. \] 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 $\Gamma_0(N)$ with nebentypus, and this applies to $t$-core partitions and generalized Frobenius partitions with $h$-colors.