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arXiv · 1407.7285

An explicit approach to the Ahlgren-Ono conjecture

Geoffrey D. Smith·Lynnelle Ye
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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 Bellaïche and Khare generalizing Serre's results on the local nilpotency of the Hecke algebra.

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Geoffrey D. Smith, Lynnelle Ye. 2014-07-27. An explicit approach to the Ahlgren-Ono conjecture. https://arxiv.org/abs/1407.7285

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