2016/01/21 by Frank Garvan, Garvan, Frank
Mathematics · #05A19 #11B65 #11F11 #11F37 #11P83 #33D15 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:05A19 #msc:11B65 #msc:11F11 #msc:11F37 #msc:11P83 #msc:33D15
paper · pdf · doi:10.48550/arxiv.1601.05781
I have corrected some typos from the previous version and added and clarified some material. I thank Chris Jennings-Shaffer, Eric Mortenson and Nick Andersen for their comments. Any additional comments or corrections are welcome
arxiv created 2016/09/08 · arxiv updated 2016/09/09
At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of R(ζ,q), where R(z,q) is the two-variable generating function of Dyson's rank function and ζ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon and McIntosh, and motivated by Dyson's question, Bringmann, Ono and Rhoades studied transformation properties of R(ζ,q). In this paper we strengthen and extend the results of Bringmann, Rhoades and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo 5 from the Lost Notebook has an analogue for all primes greater than 3. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod 7.