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 $p\ge 5$. 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 $p(5n+4)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}5)$ and $p(7n+5)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}7)$.