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.
Qihang Sun (Mon,) studied this question.