Ramanujan’s Radial Limits

Ramanujan’s famous deathbed letter to G. H. Hardy concerns the asymptotic properties of modular forms and his so-called mock theta functions. For his mock theta function f(q), he asserts, as q approaches an even order 2k root of unity, that we have f(q)− (−1)(1− q)(1− q)(1− q) · · · ` 1− 2q + 2q − · · · ́ = O(1). We give two proofs of this claim by offering exact formulas for these limiting values. One formula is a specialization of a general result which relates Dyson’s rank mock theta function and the Andrews-Garvan crank modular form. The second formula is ad hoc, and it relies on the elementary manipulation of qseries which are themselves mock modular forms. Both proofs show that the O(1) constants are not mysterious; they are values of special q-series which are finite sums at even order roots of unity.