Congruences of the partition function
Journal
International Mathematics Research Notices
Journal Volume
2011
Journal Issue
14
Pages
3261-3288
Date Issued
2011
Author(s)
Abstract
Dedicated to Professor B. C. Berndt on the occasion of his 70th birthday ABSTRACT. Let p(n) denote the partition function. In this article, we will show that congruences of the form p(m j ℓ k n + 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 j ℓ k n + B) ≡ 0 holds for all integers n not divisible by ℓ. mod m i 1.
Type
journal article