Distribution of the partition function modulo m

2000/08/17 by Ken Ono · 3 citations
Mathematics · #math.NT #msc:11P83

paper · pdf

published as Ann. of Math. (2) 151 (2000), no. 1, 293-307 · 15 pages

arxiv created 2000/08/17 · arxiv updated 2009/11/30

Abstract

Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove that there are infinitely many such congruences for every prime modulus exceeding 3. In addition, we provide a simple criterion guaranteeing the truth of Newman's conjecture for any prime modulus exceeding 3 (recall that Newman's conjecture asserts that the partition function hits every residue class modulo a given integer M infinitely often).

Cited by

• An Analogue of the Dedekind Eta Function for Hecke Groups H(√(D))
• Scarcity of partition congruences on semiprime progressions
• Rook decomposition of the Partition function

Related

• Distribution of the k-regular partition function modulo composite integers M
• Scarcity of congruences for the partition function
• Infinite Families of Congruences Modulo 5 for Ramanujan's General Partition Function
• Two partition functions with congruences modulo 3, 5, 7, and 13
• The partition function modulo 4