On three theorems of Folsom, Ono and Rhoades

Author:
Wadim Zudilin

Journal:
Proc. Amer. Math. Soc. **143** (2015), 1471-1476

MSC (2010):
Primary 11F03; Secondary 11P84, 33D15

Published electronically:
November 4, 2014

MathSciNet review:
3314062

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 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.

**[1]**George E. Andrews and Bruce C. Berndt,*Ramanujan’s lost notebook. Part I*, Springer, New York, 2005. MR**2135178****[2]**George E. Andrews and Bruce C. Berndt,*Ramanujan’s lost notebook. Part II*, Springer, New York, 2009. MR**2474043**- [3]
Bruce
C. Berndt and Robert
A. Rankin,
*Ramanujan*, History of Mathematics, vol. 9, American Mathematical Society, Providence, RI; London Mathematical Society, London, 1995. Letters and commentary. MR**1353909** **[4]**Kathrin Bringmann, Ken Ono, and Robert C. Rhoades,*Eulerian series as modular forms*, J. Amer. Math. Soc.**21**(2008), no. 4, 1085–1104. MR**2425181**, 10.1090/S0894-0347-07-00587-5**[5]**Michael Griffin, Ken Ono, and Larry Rolen,*Ramanujan’s mock theta functions*, Proc. Natl. Acad. Sci. USA**110**(2013), no. 15, 5765–5768. MR**3065809**, 10.1073/pnas.1300345110- [6]
Amanda
Folsom, Ken
Ono, and Robert
C. Rhoades,
*Mock theta functions and quantum modular forms*, Forum Math. Pi**1**(2013), e2, 27. MR**3141412** - [7]
A. Folsom, K. Ono and R.C. Rhoades,
*Ramanujan's radial limits*,*Preprint*(2012), 12 pp. - [8]
E. Mortenson,
*Eulerian series as modular forms revisited*,*Preprint*`arXiv:1304.4012 [math.NT]`(2013), 6 pp. **[9]**Ken Ono,*Unearthing the visions of a master: harmonic Maass forms and number theory*, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 347–454. MR**2555930****[10]**Yu. A. Pupyrev,*On the linear and algebraic independence of 𝑞-zeta values*, Mat. Zametki**78**(2005), no. 4, 608–613 (Russian, with Russian summary); English transl., Math. Notes**78**(2005), no. 3-4, 563–568. MR**2226733**, 10.1007/s11006-005-0155-3**[11]**Don Zagier,*Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann)*, Astérisque**326**(2009), Exp. No. 986, vii–viii, 143–164 (2010). Séminaire Bourbaki. Vol. 2007/2008. MR**2605321**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
11F03,
11P84,
33D15

Retrieve articles in all journals with MSC (2010): 11F03, 11P84, 33D15

Additional Information

**Wadim Zudilin**

Affiliation:
School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan NSW 2308, Australia

Email:
wadim.zudilin@newcastle.edu.au

DOI:
https://doi.org/10.1090/S0002-9939-2014-12364-1

Received by editor(s):
September 11, 2013

Published electronically:
November 4, 2014

Additional Notes:
The author was supported by the Australian Research Council.

Communicated by:
Ken Ono

Article copyright:
© Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.