Dyson’s ranks and Maass forms

Bringmann, Kathrin; Ono, Ken

doi:10.4007/annals.2010.171.419

Cited by 174 publications

(171 citation statements)

References 33 publications

(34 reference statements)

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“…Zwegers's thesis has sparked a flurry of recent activity involving such Maass forms. Indeed, harmonic Maass forms are now known to play a central role in the study of Ramanujan's mock theta functions, as well as other important mathematical topics: Borcherds products, derivatives of modular L-functions, GrossZagier formulas and Faltings heights of CM cycles, partitions,and traces of singular moduli (see Bringmann and Ono [5;6], Bringmann, Ono and Rhoades [7], Bruinier [8], Bruinier and Funke [9], Bruinier and Ono [10], Bruinier and Yang [12], Ono [40], Zagier [55] and Zwegers [56]). …”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

SO(3)–Donaldson invariants of ℂP²and mock theta functions

Malmendier

Ono

2012

Geom. Topol.

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We compute the Moore-Witten regularized u-plane integral on CP 2 , and we confirm the conjecture that it is the generating function for the SO.3/-Donaldson invariants of CP 2 . We also derive generating functions for the SO.3/-Donaldson invariants with 2N f massless monopoles using the geometry of certain rational elliptic surfaces (N f 2 f0; 2; 3; 4g), and we show that the partition function for N f D 4 is nearly modular. Our results rely heavily on the theory of mock theta functions and harmonic Maass forms (for example, see Ono [40]). 57R57

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

SO(3)–Donaldson invariants of ℂP²and mock theta functions

Malmendier

Ono

2012

Geom. Topol.

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“…This is contrary to Ramanujan's claim in his deathbed letter that, "Unlike the False ϑ-functions (studied partially by Prof. Rogers in his interesting paper) they ('Mock' ϑ-functions) enter into mathematics as beautifully as the ordinary theta functions." (4). The connection between partial and mock theta functions in different half-planes has appeared in at least three other contexts.…”

Section: The Mordell Integralmentioning

confidence: 99%

“…The most significant applications of mock theta functions to the theory of partitions come from the study of Dyson's rank (see the works of Bringmann and Ono [4,5]). An integer partition of n is a sequence λ 1 ≥ λ 2 ≥ · · · ≥ λ > 0 such that λ 1 +λ 2 +· · ·+λ = n. The numbers λ j are the parts of the partition.…”

Section: The Mordell Integralmentioning

confidence: 99%

The Mordell integral, quantum modular forms, and mock Jacobi forms

Chern

Rhoades

2015

Res. number theory

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It is explained how the Mordell integral R e πiτ x 2 −2πzx cosh(π x) dx unifies the mock theta functions, partial (or false) theta functions, and some of Zagier's quantum modular forms. As an application, we exploit the connections between q-hypergeometric series and mock and partial theta functions to obtain finite evaluations of the Mordell integral for rational choices of τ and z. Mathematics Subject Classification: 11P55, 05A17Keywords: Jacobi forms; mock Jacobi Form; partial Jacobi theta function; False theta function; Partial theta function; Mock theta function; Mordell integral; Ramanujan The Mordell integralThe Mordell integral isThe integral was studied by Kronecker [19,20] and in the late 1800s. A few years later, the integral underwent an intensive investigation by Mordell [28,29]. He established relationships between this integral and class number formulas. Later, Siegel [37] used properties of the integral to determine the approximate functional equation of the Riemman zeta function as well as asymptotics for its first moment. Very recently, the Mordell integral has been used to efficiently compute values of the Riemann zeta function [21]. The integral also appears in the solution of the 1-dimensional heat equation, see [32] and the references therein. Andrews [1] gave an early investigation of the Mordell integral in the study of the mock theta functions of Ramanujan.The Mordell integral plays a central role in Sander Zwegers Ph.D. thesis [42] on the mock theta functions. His work shows that the Mordell integral may be viewed as the obstruction to modularity of the mock theta functions. Moreover, he showed how to reinterpret the integral in terms of a period integral of a certain weight 3/2 modular form. His thesis has paved the way for many applications of mock theta functions to the study

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“…In the weight 1/2 case, they are intimately related to the mock theta functions, a term coined by Ramanujan in his famous 1920 deathbed letter to Hardy. It took until the first decade of the twenty-first century before work by Zwegers [43], Bruinier and Funke [7] and Bringmann and Ono [5,6] established the "right" framework for these enigmatic functions of Ramanujan's, namely that of harmonic Maaß forms. Since then, there have been many applications of harmonic Maaß forms both in various fields of pure mathematics, see for instance [1,4,9,16], among many others, and mathematical physics, especially in regard to quantum black holes and wall crossing [15] as well as Mathieu and Umbral Moonshine [12,13,18,23].…”

Section: Harmonic Maaß Formsmentioning

confidence: 99%