Bilateral series and Ramanujan's radial limits

TL;DR

This paper establishes new connections between Ramanujan's 5th order mock theta functions and bilateral series, providing detailed asymptotic behavior near roots of unity and extending the understanding of their radial limits.

Contribution

It introduces a novel method linking 5th order mock theta functions to bilateral series, enabling precise analysis of their limits at roots of unity.

Findings

Derived explicit formulas for radial limits of all 5th order mock theta functions.

Connected mock theta functions to modular bilateral series.

Extended the method to other mock theta functions.

Abstract

Ramanujan's last letter to Hardy explored the asymptotic properties of modular forms, as well as those of certain interesting $q$-series which he called \emph{mock theta functions}. For his mock theta function $f(q)$, he claimed that as $q$ approaches an even order $2k$ root of unity $\zeta$, \[\lim_{q\to \zeta} \big(f(q) - (-1)^k (1-q)(1-q^3)(1-q^5)\cdots (1-2q + 2q^4 - \cdots)\big) = O(1),\] and hinted at the existence of similar statements for his other mock theta functions. Recent work of Folsom-Ono-Rhoades provides a closed formula for the implied constant in this radial limit of $f(q)$. Here, by different methods, we prove similar results for all of Ramanujan's 5th order mock theta functions. Namely, we show that each 5th order mock theta function may be related to a modular bilateral series, and exploit this connection to obtain our results. We further explore other mock theta functions to which this method can be applied.