Scarcity of partition congruences on semiprime progressions

2025/08/27 by Ahlgren, Scott, Beckwith, Olivia
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.19512

Abstract

In recent work with Raum the authors considered congruences for the ordinary partition function p(n) of the form p(ℓ Q^r n+β)≡ 0\pmodℓ where ℓ, Q≥ 5 are prime and r∈ \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∈ ℤ with p(ℓ Q n+β)≡ 0\pmod ℓ 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 ℓ.

Citations

• The Shimura lift and congruences for modular forms with the eta multiplier
• Congruences like Atkin's for the partition function
• Congruences for Level 1 cusp forms of half-integral weight
• Scarcity of congruences for the partition function
• Distribution of the partition function modulo m

Related

• Scarcity of congruences for the partition function
• Congruence relations for r-colored partitions
• Congruences like Atkin's for the partition function
• A note on odd partition numbers
• Congruences of the partition function