Scarcity of partition congruences on semiprime progressions

Abstract

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