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

TL;DR

The paper advances Newman's Conjecture for partition functions modulo integers by proving that, for any fixed number of distinct prime divisors d, almost all moduli M in B_d satisfy the conjecture across all residue classes. The authors achieve this by translating partition generating functions into modular forms of half-integral or integral weight and studying the distribution of their Fourier coefficients modulo M using Galois representations, Chebotarev density, and Treneer-type congruence constructions. The results yield density-1 statements for generalized Frobenius partitions with h colors and for t-core partitions, linking combinatorial partition statistics to deep modular-form arithmetic. The methods provide a unifying approach to derive congruence distributions and have implications for broader families of partition-like objects via eta-quotients and Shimura correspondences.

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 $Γ_0(N)$ with nebentypus, and this applies to $t$-core partitions and generalized Frobenius partitions with $h$-colors.

Theorems & Definitions (24)

• Conjecture 1.1: Newman's Conjecture
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• Remark 2.2