A new approach to the Dyson rank conjectures

TL;DR

This paper introduces a more elementary method using Hecke-Rogers series to analyze Dyson's rank conjectures, providing an alternative to previous proofs based on elliptic functions and harmonic Maass forms.

Contribution

The paper presents a novel, elementary approach to Dyson's rank conjectures employing Hecke-Rogers series, simplifying the proof process.

Findings

New proof of Dyson's rank conjectures using Hecke-Rogers series

Simplification of existing proofs by avoiding elliptic functions

Potential for broader applications in partition theory

Abstract

In 1944 Dyson defined the rank of a partition as the largest part minus the number of parts, and conjectured that the residue of the rank mod 5 divides the partitions of 5n+4 into five equal classes. This gave a combinatorial explanation of Ramanujan's famous partition congruence mod 5. He made an analogous conjecture for the rank mod 7 and the partitions of 7n+5. In 1954 Atkin and Swinnerton-Dyer proved Dyson's rank conjectures by constructing several Lambert-series identities basically using the theory of elliptic functions. In 2016 the author gave another proof using the theory of weak harmonic Maass forms. In this paper we describe a new and more elementary approach using Hecke-Rogers series.