Vanishing properties of Kloosterman sums and Dyson's conjectures

TL;DR

This paper proves special vanishing properties of Kloosterman sums for primes 5 and 7, leading to a new proof of Dyson's rank conjectures and Ramanujan's partition congruences.

Contribution

It establishes vanishing properties of Kloosterman sums for specific primes, providing a novel proof of Dyson's rank conjectures and related partition congruences.

Findings

Vanishing properties of Kloosterman sums for p=5 and 7

New proof of Dyson's rank conjectures

Alternative proof of Ramanujan's partition congruences

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