  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 an
upcoming paper, we will provide a new proof of Dyson's conjectures by showing
that the certain Kloosterman sums vanish.
