Dyson’s ranks and Appell–Lerch sums

Denote by p ( n ) the number of partitions of n and by N ( a , M ; n ) the number of partitions of n with rank congruent to a modulo M . We find and prove a general formula for Dyson’s ranks by considering the deviation of the ranks from the average: D ( a , M ) : = ∑ n = 0 ∞ N ( a , M ; n ) - p ( n ) M q n . Using Appell–Lerch sum properties we decompose D ( a , M ) into modular and mock modular parts so that the mock modular component is supported on certain arithmetic progressions, whose modulus we can control. Using our decomposition, we show how our formula gives as a straightforward consequence Atkin and Swinnerton-Dyer’s results on ranks as well as celebrated work of Bringmann, Ono, and Rhoades’s on Maass forms. We also give an example of how our methods apply to work of Lovejoy and Osburn on rank differences.