On three theorems of Folsom, Ono and Rhoades

TL;DR

This paper provides an elementary proof of Ramanujan's original claim about mock modular functions and discusses heuristics that could lead to a new proof of a recent theorem by Folsom, Ono, and Rhoades related to mock theta and modular functions.

Contribution

It offers a new elementary proof of Ramanujan's claim and suggests heuristics for proving the Folsom--Ono--Rhoades theorem.

Findings

Elementary proof of Ramanujan's mock modular function claim

Heuristics for proving Folsom--Ono--Rhoades theorem

Potential framework for future proofs of related theorems

Abstract

In his deathbed letter to Hardy, Ramanujan gave a vague definition of a mock modular function: at each root of unity its asymptotics matches the one of a modular form, though a choice of the modular function depends on the root of unity. Recently Folsom, Ono and Rhoades have proved an elegant result about the match for a general family related to Dyson's rank (mock theta) function and the Andrews--Garvan crank (modular) function---the match with explicit formulae for implied $O(1)$ constants. In this note we give another elementary proof of Ramanujan's original claim and outline some heuristics which may be useful for obtaining a new proof of the general Folsom--Ono--Rhoades theorem.