Asymptotic expansions, \(L\)-values and a new quantum modular form. (English) Zbl 1370.11066

Summary: In [Clay Math. Proc. 11, 659–675 (2010; Zbl 1294.11084)], D. Zagier introduced the notion of a quantum modular form. One of his first examples was the “strange” function \(F(q)\) of Kontsevich. Here we produce a new example of a quantum modular form by making use of some of Ramanujan’s mock theta functions. Using these functions and their transformation behaviour, we also compute asymptotic expansions similar to expansions of \(F(q)\).


11F99 Discontinuous groups and automorphic forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols


Zbl 1294.11084
Full Text: DOI arXiv


[1] Bringmann, K., Folsom, A.: On the asymptotic behavior of Kac-Wakimoto characters. Proc. Am. Math. Soc. 141(5), 1567-1576 (2013) · Zbl 1277.11035
[2] Chern, B., Rhoades, R.C.: The Mordell integral and quantum modular forms (2012, preprint) · Zbl 1378.11058
[3] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. III. Krieger, Melbourne (1981). Based on notes left by Harry Bateman. Reprint of the 1955 original · Zbl 0064.06302
[4] Folsom, A., Ono, K., Rhoades, R.C.: Ramanujan’s radial limits (2012, submitted for publication) · Zbl 1359.11064
[5] Gordon, B.; McIntosh, R.; Alladi, K. (ed.); Garvan, F. (ed.), A survey of classical mock theta functions, No. 23, 95-144 (2012), New York · Zbl 1246.33006
[6] Watson, G.N.: The final problem: an account of the Mock theta functions. J. Lond. Math. Soc. S1-11(1), 55 (1936) · Zbl 0013.11502
[7] Zagier, D.: Vassiliev invariants and a strange identity related to the Dedekind eta-function. Topology 40(5), 945-960 (2001) · Zbl 0989.57009
[8] Zagier, D., Quantum modular forms, No. 11, 659-675 (2010), Providence · Zbl 1294.11084
[9] Zwegers, S.P.: Mock theta functions. Ph.D. Thesis (Advisor: D. Zagier), Universiteit Utrecht (2002) · Zbl 1194.11058
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