Bilateral series and Ramanujan’s radial limits

Published

Journal Article

© 2014 American Mathematical Society. limq→ζ (f(q) − (−1)k(1 − q)(1 − q3)(1 − q5) · · · (1 − 2q + 2q4 − ·· ·)) = O(1),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 mock theta functions. For his mock theta function f(q), he claimed that as q approaches an even order 2k root of unity ζ,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.

Full Text

Cited Authors

  • Bajpai, J; Kimport, S; Liang, J; Ma, D; Ricci, J

Published Date

  • January 1, 2015

Published In

Volume / Issue

  • 143 / 2

Start / End Page

  • 479 - 492

Electronic International Standard Serial Number (EISSN)

  • 1088-6826

International Standard Serial Number (ISSN)

  • 0002-9939

Digital Object Identifier (DOI)

  • 10.1090/S0002-9939-2014-12249-0

Citation Source

  • Scopus