The partition function modulo 3 in arithmetic progressions

Abstract

Let \(p(n)\) be the partition function. Ahlgren and Ono conjectured that every arithmetic progression contains infinitely many integers \(N\) for which \(p(N)\) is not congruent to \(0\;(\mathrm{mod}\;3)\). Radu proved this conjecture in 2010 using the work of Deligne and Rapoport. In this note, we give a simpler proof of Ahlgren and Ono’s conjecture in the special case where the modulus of the arithmetic progression is a power of \(3\) by applying a method of Boylan and Ono and using the work of Bellaiche and Khare generalizing Nicolas and Serre’s results on the local nilpotency of the Hecke algebra.