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Title: 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(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) equivalent to r (mod M). For a positive integer d, let Bd 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 Bd 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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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