Exact formulae for ranks of partitions

TL;DR

This paper derives exact formulas for the ranks of partitions using advanced number theory techniques, extending previous asymptotic results to precise, finite expressions involving Kloosterman sums, especially for prime moduli.

Contribution

It proves that Bringmann's asymptotic formula yields a Rademacher-type exact formula for partition ranks at prime moduli, involving vector-valued Kloosterman sums.

Findings

Derivation of exact formulas for partition ranks.

Connection between asymptotic formulas and Rademacher-type sums.

Implication for proving Dyson's conjectures using Kloosterman sums.

Abstract

In 2009, Bringmann arXiv:0708.0691 [math.NT] used the circle method to prove an asymptotic formula for the Fourier coefficients of rank generating functions. In this paper, we prove that Bringmann's formula, when summing up to infinity and in the case of prime modulus, gives a Rademacher-type exact formula involving sums of vector-valued Kloosterman sums. As a corollary, in another paper arXiv:2406.07469 [math.NT], we will provide a new proof of Dyson's conjectures by showing that the certain Kloosterman sums vanish.