Vanishing properties of Kloosterman sums and Dyson’s conjectures
Copyright policy )Vanishing properties of Kloosterman sums and Dyson’s conjectures
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 $p(7n+5)\equiv 0\pmod 7$.
Partitions; congruences and congruential restrictions, Dyson's rank, Mathematics - Number Theory, Rademacher exact formula, 11L05, 11P83, 11P82, FOS: Mathematics, Analytic theory of partitions, Kloosterman sum, Number Theory (math.NT), Gauss and Kloosterman sums; generalizations, integer partition
Partitions; congruences and congruential restrictions, Dyson's rank, Mathematics - Number Theory, Rademacher exact formula, 11L05, 11P83, 11P82, FOS: Mathematics, Analytic theory of partitions, Kloosterman sum, Number Theory (math.NT), Gauss and Kloosterman sums; generalizations, integer partition
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