Congruences of the partition function

2009/04/16 by Yifan Yang, Yang, Yifan
Mathematics · #11F25 #11F37 #11P82 #11P83 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0904.2530

openalex publication_date 2009/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p(n) denote the partition function. In this article, we will show that congruences of the form p(m^jℓ^kn+B)≡ 0\mod m for all n≥ 0 exist for all primes m and ℓ satisfying m≥ 13 and ℓ≠ 2,3,m. Here the integer k depends on the Hecke eigenvalues of a certain invariant subspace of S_m/2-1(Γ₀(576),χ₁₂) and can be explicitly computed. More generally, we will show that for each integer i>0 there exists an integer k such that for every non-negative integers j≥ i with a properly chosen B the congruence p(m^jℓ^kn+B)≡ 0\mod mⁱ holds for all integers n not divisible by ℓ.

Citations

• On Modular Forms of Half Integral Weight

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