Papers › Vanishing properties of Kloosterman sums and Dyson's conjectures
Vanishing properties of Kloosterman sums and Dyson's conjectures
Qihang Sun
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In a previous paper arXiv:2406.06294 [math.NT], the author proved the exact formulae for ranks of partitions modulo each prime p≥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)≡0pmod5 and p(7n+5)≡0pmod7.
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