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 $p(\ell Q^{r}n+\beta )\equiv 0(\mathrm{mod}\ell )$ where $\ell ,Q\ge 5$ 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 $p(\ell Qn+\beta )\equiv 0(\mathrm{mod}\ell )$ 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 $\ell$.