Scarcity of partition congruences on semiprime progressions

TL;DR

This work addresses the scarcity of partition congruences of the form $p(\ell Q n+\beta)\equiv 0\pmod{\ell}$ with $r=1$ and primes $\ell,Q\ge 5$. It develops a framework based on half-integral weight modular forms with the eta multiplier and leverages a recent mod $\ell$ level-one result of Dicks, together with Radu’s square-class propagation, to show that, outside trivial Ramanujan-type cases, the set of primes $Q$ admitting such a congruence for some $\beta$ has density zero. The main contribution is a refined argument that excludes the possibility of congruences supported on finitely many square classes, thereby proving density zero for almost all $Q$ and establishing a strong scarcity result for these partition congruences. This advances the understanding of congruences for $p(n)$ in semiprime progressions and demonstrates the power of mod $\ell$ modular-form techniques in ruling out broad families of arithmetic progressions.

Abstract

In recent work with Raum the authors considered congruences for the ordinary partition function $p(n)$ of the form $p(\ell Q^r n+β)\equiv 0\pmod\ell$ where $\ell, Q\geq 5$ are prime and $r\in \{1,2\}$, and proved a number of results which show that such congruences are scarce in a precise sense. Here we improve one of our results when $r=1$; in particular we prove (outside of trivial cases) that the set of primes $Q$ such that there exists $β\in \mathbb{Z}$ with $p(\ell Q n+β)\equiv 0\pmod \ell$ for all $n$ has density zero. The proof involves a modification of part of our previous argument and an application of a recent theorem of Dicks regarding modular forms of half-integral weight and level one modulo $\ell$.

Theorems & Definitions (6)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Proposition 3.1