Vanishing properties of Kloosterman sums and Dyson's conjectures

TL;DR

The paper develops vanishing properties for vector-valued Kloosterman sums that appear in exact formulae for partition ranks modulo primes p=5,7. By a detailed case analysis based on gcd structure of $c/p$ and the construction of $V(r,c)$, it shows cancellations among Kloosterman-sum contributions, yielding new proofs of Dyson's rank conjectures and Ramanujan congruences. The method provides a conceptually different route from previous modular-form approaches, using explicit arithmetic of Dedekind sums and Arg-difference configurations to force vanishing. The results thus give streamlined proofs of $p(5n+4) ≡ 0$ mod 5 and $p(7n+5) ≡ 0$ mod 7 and reinforce the link between rank statistics and modular objects.

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

Theorems & Definitions (15)

• Theorem 1.1: AtkinSDrank
• Remark
• Theorem 1.2: QihangExactFormula
• Theorem 1.3
• Corollary 1.4
• Proposition 3.1
• Remark
• Lemma 3.3
• proof : Proof of Lemma \ref{['p=5 Argreduction']}
• proof : Proof of Proposition \ref{['Vrc sum equals 0']}