Dyson's Ranks and Appell-Lerch Sums

Abstract

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: \begin{equation*} D(a,M) := \sum_{n= 0}^{\infty}\left(N(a,M;n) - \frac{p(n)}{M}\right) q^n. \end{equation*} 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 Bringmann, Ono, and Rhoades's results on Maass forms. We also apply our techniques to a variation of Dyson's ranks due to Berkovitch and Garvan.

substantially revised

References in corpus (2)

• Mock Theta Functions
• Eulerian series as modular forms revisited

Cited by in corpus (7)

• On ranks and cranks of partitions modulo $4$ and $8$
• On the tenth-order mock theta functions
• Proofs of Some Conjectures of Chan on Appell-Lerch Sums
• Deviation of the rank and crank modulo 11
• Splitting Appell functions in terms of single quotients of theta functions
• Short proofs of Ramanujan-like identities for the eighth order mock theta function $V_0(q)$
• Rank deviations for overpartitions