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arXiv · 2406.07469

Vanishing properties of Kloosterman sums and Dyson's conjectures

Qihang Sun
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Abstract

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 $p(7n+5)\equiv 0\pmod 7$.

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Qihang Sun. 2024-06-11. Vanishing properties of Kloosterman sums and Dyson's conjectures. https://doi.org/10.1007/s11139-024-01011-4

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