Vanishing properties of Kloosterman sums and Dyson's conjectures
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916650498916352 |
|---|---|
| author | Sun, Qihang |
| author_facet | Sun, Qihang |
| contents | In a previous paper arXiv:2406.06294 [math.NT], the author proved the exact formulae for ranks of partitions modulo each prime $p\geq 5$. In this paper, for $p=5$ and $7$, we prove special vanishing properties of the Kloosterman sums appearing in the exact formulae. These vanishing properties imply a new proof of Dyson's rank conjectures. Specifically, we give a new proof of Ramanujan's congruences $p(5n+4)\equiv 0\pmod 5$ and $p(7n+5)\equiv 0\pmod 7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07469 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vanishing properties of Kloosterman sums and Dyson's conjectures Sun, Qihang Number Theory 11L05, 11P83, 11P82 In a previous paper arXiv:2406.06294 [math.NT], the author proved the exact formulae for ranks of partitions modulo each prime $p\geq 5$. In this paper, for $p=5$ and $7$, we prove special vanishing properties of the Kloosterman sums appearing in the exact formulae. These vanishing properties imply a new proof of Dyson's rank conjectures. Specifically, we give a new proof of Ramanujan's congruences $p(5n+4)\equiv 0\pmod 5$ and $p(7n+5)\equiv 0\pmod 7$. |
| title | Vanishing properties of Kloosterman sums and Dyson's conjectures |
| topic | Number Theory 11L05, 11P83, 11P82 |
| url | https://arxiv.org/abs/2406.07469 |