Dyson’s ranks and Maass forms

By Kathrin Bringmann, Ken Ichi Ono. 2010

234 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/3ff99b4689d1c2e554c7f4bdcfb7ec0d16fee641

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Rank and crank moments for overpartitions (2008, background): “As in the proof of Proposition 4.1 of [ 9 ], one can show that the non-holomorphic parts ofM1( ) and M( ) are supported on negative squares.”
Asymptotic inequalities for positive crank and rank moments (2012, background): “However, the study of these statistics does not end with the Ramanujan congruences, as there has been a great wealth of further results regarding the arithmetic properties of the crank and rank [13, 24, 27], their connections to modular and automorphic forms, including mock theta functions [14, 19],”
An extension of the Hardy-Ramanujan circle method and applications to partitions without sequences (2011, background): “Instead, recent works of the first author, Ono, and Zwegers [10, 12, 25] show that f(q) is best understood as the holomorphic part of a harmonic Maass form of half-integral weight.”
ℓ-Adic properties of the partition function☆☆☆ (2012, background): “Indeed, the subject of ranks and cranks represents a different thread in number theory which has grown out of the problem of trying to better understand partition congruences (for example, see [5, 11, 13, 14, 15, 17, 20, 23] to name a few).”
Hecke‐type double sums, Appell–Lerch sums, and mock theta functions, I (2012, background): “Zwegers thus established the modularity theory for mock theta functions, and his work allows mock theta functions to be cast as the holomorphic parts of weak Maass forms [11, 30, 9, 10]).”
Unearthing the visions of a master: harmonic Maass forms and number theory (2008, background): “Using a new class of harmonic Maass forms, which will be described in Section 8.4, the author and K. Bringmann [47] [55] have obtained general results in this direction.”
Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues (2008, background): “These results are related to the vanishing of derivatives of quadratic twists of weight 2 modular L-functions at s = 1. v) It would be interesting to find an explicit construction of good harmonic weak Maass forms for CM newforms.”
Umbral Moonshine (2012, background): “Interest in and applications of mock theta functions has burgeoned during the last decade following the work of Zwegers [44] who found an intrinsic definition of mock theta functions and their near modular behavior, and many applications of his work can be found in combinatorics [46, 47], characters”
Mock Jacobi forms in basic hypergeometric series (2008, background): “In his ground-breaking work on mock theta functions [19], Zwegers constructed a non-holomorphic but real analytic Jacobi form whose holomorphic part is a normalized Lerch sum and whose nonholomorphic part can be written in terms of a period integral of a weight 3/2 unary theta series.”
Asymptotics for rank and crank moments (2009, background): “As a result, the study of generating functions from partition theory has inspired a number of important results about mock theta functions and harmonic Maass forms [9, 12, 14, 15].”
Inequalities between ranks and cranks (2009, methodology): “This has allowed the use of techniques taken from combinatorics, q-series, and complex analysis, and has led to a number of important results about harmonic Maass forms and mock theta functions [8, 10, 11, 12 ].”
Zagier-type dualities and lifting maps for harmonic Maass–Jacobi forms (2010, background): “His significant discovery has led to major applications in different areas of mathematics and physics, such as Bringmann and Ono’s [9,6] solutions of well-known conjectures in combinatorics and the theory of q-series.”
RANK AND CONGRUENCES FOR OVERPARTITION PAIRS (2008, background): “Instead of asking if the rank implies congruences in the sense of Theorem 1.2, we may ask if congruences for pp(n) are implied by the fact that the counting functions NN(r, t, An + B) themselves satisfy congruences.”
Taylor coefficients of mock-Jacobi forms and moments of partition statistics (2014, background): “In this case, if the elliptic variable of one of Zwegers’ functions is specialized to a root of unity, then the resulting holomorphic part is a mock modular form, which is now known to be related to the theory of harmonic weak Maass forms (see [16] for more on the development of this theory, and [13”
MOCK THETA FUNCTIONS AND QUANTUM MODULAR FORMS (2013, methodology): “We provide an explicit closed formula for the implied constants in Ramanujan’s claim, and show that they are 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, two of the most studied ”