An explicit approach to the Ahlgren-Ono conjecture

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 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 work of Bellaïche and Khare generalizing Serre's results on the local nilpotency of the Hecke algebra.

5 pages

References in corpus (2)

• Formes modulaires modulo 2 : l'ordre de nilpotence des opérateurs de Hecke
• Formes modulaires modulo 2 : structure de l'algèbre de Hecke