Exact formulae for ranks of partitions

TL;DR

The paper proves that Bringmann’s circle-method asymptotics for rank generating coefficients $A(\frac{\ell}{p};n)$, with prime modulus $p$, can be upgraded to an exact Rademacher-type formula expressed as sums of vector-valued Kloosterman sums, integrated against $I_{1/2}$-Bessel-type terms. It develops a full scalar and vector-valued automorphic framework, including multiplier systems, Maass and harmonic Maass forms, and Maass–Poincaré series, and then constructs principal parts to force an exact, convergent expansion. This yields a concrete exact formula for rank coefficients and aligns with Bringmann’s asymptotics when summed to infinity; as a corollary, Dyson’s conjectures follow from vanishing properties of certain vector-valued Kloosterman sums. The methods bridge partition rank theory with the spectral theory of vector-valued automorphic forms and provide a robust toolkit for exact rank formulas at prime moduli.

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.

Theorems & Definitions (39)

• Theorem 1.1: BrmOno2006ivt
• Theorem 1.2: QihangFirstAsympt
• Theorem 1.3: BringmannTAMS
• Theorem 1.4
• Remark
• Theorem 2.1: AtkinSDrank
• Remark
• Corollary 2.2
• proof
• Theorem 2.3: QihangVanishingKLsums2025