TY - JOUR

T1 - Dyson’s ranks and Appell–Lerch sums

AU - Hickerson, Dean

AU - Mortenson, Eric

PY - 2017/2/1

Y1 - 2017/2/1

N2 - 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: (Formula presented.) 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.

AB - 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: (Formula presented.) 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.

KW - 11B65

KW - 11F11

KW - 11F27

KW - 11P84

UR - http://www.scopus.com/inward/record.url?scp=84960355605&partnerID=8YFLogxK

U2 - 10.1007/s00208-016-1390-5

DO - 10.1007/s00208-016-1390-5

M3 - Article

AN - SCOPUS:84960355605

VL - 367

SP - 373

EP - 395

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 1-2

ER -