An explicit approach to the Ahlgren-Ono conjecture

TL;DR

This paper provides a simplified proof for a special case of the Ahlgren-Ono conjecture regarding the non-vanishing of partition function values modulo 3 within certain arithmetic progressions, using advanced algebraic methods.

Contribution

It offers a more straightforward proof for the case where the modulus is a power of 3, extending previous results with new algebraic techniques.

Findings

Confirmed the conjecture for powers of 3 as modulus

Simplified the proof approach compared to previous work

Extended understanding of partition function congruences

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 $0\pmod{3}$. 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 Bella\"iche and Khare generalizing Serre's results on the local nilpotency of the Hecke algebra.