On three theorems of Folsom, Ono and Rhoades

Abstract:

In his deathbed letter to G. H. Hardy, Ramanujan gave a vague definition of a mock modular function: at each root of unity its asymptotic 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.