Ramanujan’s mock theta functions

Griffin, Michael; Ono, Ken; Rolen, Larry

doi:10.1073/pnas.1300345110

Cited by 27 publications

(22 citation statements)

References 13 publications

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“…Griffin, the second author, and Rolen [28] have confirmed Ramanujan's speculation. There are no weakly holomorphic modular forms which exactly cut out the singularities of Ramanujan's mock theta functions.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 58%

Mock Theta Functions and Quantum Modular Forms

Folsom¹,

Ono²,

Rhoades³

2013

Forum math. Pi

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114

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Ramanujan's last letter to Hardy concerns the asymptotic properties of modular forms and his 'mock theta functions'. For the mock theta function f (q), Ramanujan claims that as q approaches an even-order 2k root of unity, we have f (q) − (−1) k (1 − q)(1 − q 3 )(1 − q 5 ) · · · (1 − 2q + 2q 4 − · · ·) = O(1).We prove Ramanujan's claim as a special case of a more general result. The implied constants in Ramanujan's claim are not mysterious. They arise in Zagier's theory of 'quantum modular forms'. We provide explicit closed expressions for these 'radial limits' as values of a 'quantum' q-hypergeometric function which underlies a new relationship between Dyson's rank mock theta function and the Andrews-Garvan crank modular form. Along these lines, we show that the Rogers-Fine false ϑ-functions, functions which have not been well understood within the theory of modular forms, specialize to quantum modular forms. 2010 Mathematics Subject Classification: 11F99 (primary); 11F37, 33D15 (secondary) Overview In his 1920 deathbed letter to Hardy, Ramanujan gave examples of 17 curious q-series he referred to as 'mock theta functions' [11]. In the decades following Ramanujan's death, mathematicians were unable to determine how these functions fit into the theory of modular forms, despite c The Author(s) 2013. The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence . A. Folsom, K. Ono, and R. C. Rhoades 2 their rather ubiquitous nature. Finally, the 2002 Ph.D. thesis of Zwegers [45] showed that while the mock theta functions were not modular, they could be 'completed' to produce real analytic vector-valued modular forms. Zwegers's breakthrough catalyzed the development of the overarching theory of 'weak Maass forms' by Bringmann, Ono, and collaborators [15, 16, 35, 42]. Ramanujan's mock theta functions, it turns out, are examples of 'holomorphic parts' of weak Maass forms, originally defined by Bruinier and Funke [18].While the theory of weak Maass forms has led to a flood of applications in many disparate areas of mathematics (see [35,42] and references therein), it is still not the case that we fully understand the deeper framework surrounding the contents of Ramanujan's last letter to Hardy. Here, we revisit Ramanujan's original claims and motivations. His last letter summarizes asymptotic properties near roots of unity of modular 'Eulerian' series. Ramanujan asks whether other Eulerian series with similar asymptotics are necessarily the sum of a modular theta function and a function which is O(1) at all roots of unity. He writes: 'The answer is it is not necessarily so . . . I have not proved rigorously that it is not necessarily so . . . But I have constructed a number of examples . . . '. In fact, Ramanujan's sole example and claim pertain to his third-order mock theta function f (q). CLAIM (Ramanujan [11]). As q approaches an even-order 2k root of unity, we haveHere, we prove (in ...

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Section: Introduction and Statement Of Resultsmentioning

confidence: 58%

Mock Theta Functions and Quantum Modular Forms

Folsom¹,

Ono²,

Rhoades³

2013

Forum math. Pi

Self Cite

114

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“…They, together with Sutherland, conjectured that the associated polynomials always generate ring class fields and are irreducible ( [6], p. 20), which was shown by Mertens and Rolen in [17]. Many others have studied properties of these non-holomorphic singular moduli (see [1,7,12,13,14]). Here, we consider the general problem of constructing class invariants (i.e., modular forms whose CM-values generate Hilbert class fields) from non-holomorphic modular forms of a special type.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Class invariants for certain non-holomorphic modular functions

2015

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“…Griffin, Ono, and Rolen [10] proved that all of Ramanujan's examples satisfy Definition 1.1. Rhoades [13] asked if one can choose the set {M ζ (q)} ζ to be finite and can also give explicit expressions for the radial limits in (ii).…”

Section: Introductionmentioning

confidence: 99%

Radial limits of the universal mock theta function 𝑔₃

Jang

Löbrich

2016

Proc. Amer. Math. Soc.

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Referring to Ramanujan's original definition of a mock theta function, Rhoades asked for explicit formulas for radial limits of the universal mock theta functions g 2 and g 3 . Recently, Bringmann and Rolen found such formulas for specializations of g 2 . Here we treat the case of g 3 , generalizing radial limit results for the rank generating function of Folsom, Ono, and Rhoades. Furthermore, we find expressions for radial limits of fifth order mock theta functions different from those of Bajpai, Kimport, Liang, Ma, and Ricci.

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Ramanujan’s mock theta functions

Cited by 27 publications

References 13 publications

Mock Theta Functions and Quantum Modular Forms

Mock Theta Functions and Quantum Modular Forms

Class invariants for certain non-holomorphic modular functions

Radial limits of the universal mock theta function 𝑔₃

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