2020/06/13 by Scott Ahlgren, Ahlgren, Scott, Olivia Beckwith +3 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2006.07645
openalex publication_date 2020/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The arithmetic properties of the ordinary partition function p(n) have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form p(ℓ n+β)≡ 0\pmodℓ for the primes ℓ=5, 7, 11, and it is known that there are no others of this form. On the other hand, for every prime ℓ≥ 5 there are infinitely many examples of congruences of the form p(ℓ Qm n+β)≡ 0\pmodℓ where Q≥ 5 is prime and m≥ 3. This leaves open the question of the existence of such congruences when m=1 or m=2 (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve X(ℓ Q), Galois representations and the arithmetic large sieve.