Scarcity of partition congruences on semiprime progressions

TL;DR

This paper investigates the rarity of certain partition congruences involving primes and shows that such congruences are extremely scarce, with the set of primes satisfying these conditions having density zero.

Contribution

The authors improve previous results by proving the set of primes Q for which specific partition congruences hold has density zero, using modular forms and recent theorems.

Findings

The set of primes Q with such congruences has density zero.

The proof involves a modification of previous arguments and recent modular form theorems.

The results show the scarcity of these partition congruences outside trivial cases.

Abstract

In recent work with Raum the authors considered congruences for the ordinary partition function $p(n)$ of the form $p(\ell Q^r n+\beta)\equiv 0\pmod\ell$ 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 $\beta\in \mathbb{Z}$ with $p(\ell Q n+\beta)\equiv 0\pmod \ell$ 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 $\ell$.