Exact formulae for ranks of partitions

In 2009, Bringmann [Trans. Amer. Math. Soc. 361 (2009), pp. 3483–3500] 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 [Ramanujan J. 66 (2025)], we will provide a new proof of Dyson’s conjectures by showing that the certain Kloosterman sums vanish.