Deducing the positive odd density of $p(n)$ from that of a multipartition function: An unconditional proof
| Autor: | Fabrizio Zanello |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Algebra and Number Theory
Conjecture Mathematics - Number Theory Primary: 11P83 Secondary: 05A17 11P84 11F33 Mathematics::Number Theory 010102 general mathematics 010103 numerical & computational mathematics Function (mathematics) Partition function (mathematics) 01 natural sciences Combinatorics Corollary FOS: Mathematics Mathematics - Combinatorics Multipartition Combinatorics (math.CO) Number Theory (math.NT) 0101 mathematics Mathematics |
| DOI: | 10.48550/arxiv.2103.09933 |
| Popis: | A famous conjecture of Parkin-Shanks predicts that $p(n)$ is odd with density $1/2$. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with Judge, we introduced a different approach and conjectured the "striking" fact that, if for any $A \equiv \pm 1\ (\bmod 6)$ the multipartition function $p_A(n)$ has positive odd density, then so does $p(n)$. Similarly, the positive odd density of any $p_{A}(n)$ with $A\equiv 3\ (\bmod 6)$ would imply that of $p_3(n)$. Our conjecture was shown to be a corollary of an earlier conjecture of the same paper. In this brief note, we provide an unconditional proof of it. An important tool will be Chen's recent breakthrough on a special case of our earlier conjecture. Comment: Minor updates. To appear in the J. of Number Theory |
| Databáze: | OpenAIRE |
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