Kathrin Bringmann

Eulerian series as modular forms

In 1988, Hickerson proved the celebrated “mock theta conjectures”, a collection of ten identities from Ramanujan’s “lost notebook” which express certain modular forms as linear combinations of mock theta functions. In the context of Maass forms, these identities arise from the peculiar phenomenon that two different harmonic Maass forms may have the same non-holomorphic parts. Using this perspective, we construct several infinite families of modular forms which are differences of mock theta functions.

ON THE EXPLICIT CONSTRUCTION OF HIGHER DEFORMATIONS OF PARTITION STATISTICS

The modularity of the partition generating function has many important consequences, for example asymptotics and congruences for p(n). In a series of papers the author and Ono (12, 13) connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions which are related to modular forms and which were rst considered in (11). Here we do a further step towards understanding how weak Maass forms arise from interesting partition statistics by placing certain 2-marked Durfee symbols introduced by Andrews (1) into the framework of weak Maass forms. To do this we construct a new class of functions which we call quasiweak Maass forms because they have quasimodular forms as components. As an application we prove two conjectures of Andrews. It seems that this new class of functions will play an important role in better understanding weak Maass forms of higher weight themselves, and also their derivatives. As a side product we introduce a new method which enables us to prove transformation laws for generating functions over incomplete lattices.

Lifting cusp forms to Maass forms with an application to partitions.

For 2 < k ∈ 12ℤ, we define lifts of cuspidal Poincaré series in Sk(Γ0(N)) to weight 2 − k harmonic weak Maass forms. This construction answers a question of Dyson by providing the general framework “explaining” Ramanujans mock theta functions. As an application, we show that the number of partitions of a positive integer n is the “trace” of singular moduli of a Maass form arising from the lift of a weight 4 cusp form corresponding to a Calabi–Yau threefold.

Asymptotics for rank partition functions

In this paper, we obtain asymptotic formulas for an infinite class of rank generat- ing functions. As an application, we solve a conjecture of Andrews and Lewis on inequalities between certain ranks.

Asymptotic inequalities for positive crank and rank moments

Andrews, Chan, and Kim recently introduced a modified definition of crank and rank moments for integer partitions that allows the study of both even and odd moments. In this paper, we prove the asymptotic behavior of these moments in all cases. Our main result states that the two families of moment functions are asymptotically equal, but the crank moments are also asymptotically larger than the rank moments. Andrews, Chan, and Kim also gave a combinatorial description for the differences of the first crank and rank moments that they named the ospt-function. Our main results therefore also give the asymptotic behavior of the ospt-function (and its analogs for higher moments), and we further determine the behavior of the ospt-function modulo 2 by relating its parity to Andrews’ spt-function.

Overpartitions and class numbers of binary quadratic forms

We show that the Zagier–Eisenstein series shares its nonholomorphic part with certain weak Maass forms whose holomorphic parts are generating functions for overpartition rank differences. This has a number of consequences, including exact formulas, asymptotics, and congruences for the rank differences as well as q-series identities of the mock theta type.

RANK AND CONGRUENCES FOR OVERPARTITION PAIRS

The rank of an overpartition pair is a generalization of Dysons rank of a partition. The purpose of this paper is to investigate the role that this statistic plays in the congruence properties of , the number of overpartition pairs of n. Some generating functions and identities involving this rank are also presented.

Coefficients of Harmonic Maass Forms

Harmonic Maass forms have recently been related to many different topics in number theory: Ramanujan’s mock theta functions, Dyson’s rank generating functions, Borcherds products, and central values and derivatives of quadratic twists of modular L-functions. Motivated by these connections, we obtain exact formulas for the coefficients of harmonic Maass forms of nonpositive weight, and we obtain a conditional result for such forms of weight 1 ∕ 2. This extends earlier work of Rademacher and Zuckerman in the case of weakly holomorphic modular forms of negative weight.

Asymptotics for rank and crank moments

Moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves a conjecture of two of the authors that refined a conjecture of Garvan. Garvans original conjecture states that the moments of the crank function are always larger than the moments of the rank function, even though the moments have the same main asymptotic term. The refined version provides precise asymptotic estimates for both the moments and their differences. Our proof uses the Hardy–Ramanujan Circle Method, multiple sums of Bernoulli polynomials and the theory of quasimock theta functions.

Taylor coefficients of mock-Jacobi forms and moments of partition statistics

We develop a new technique for deriving asymptotic series expansions for moments of combinatorial generating functions that uses the transformation theory of Jacobi forms and “mock” Jacobi forms, as well as the Hardy-Ramanujan Circle Method. The approach builds on a suggestion of Zagier, who observed that the moments of a combinatorial statistic can be simultaneously encoded as the Taylor coefficients of a function that transforms as a Jacobi form. Our use of Jacobi transformations is a novel development in the subject, as previous results on the asymptotic behavior of the Taylor coefficients of Jacobi forms have involved the study of each such coefficient individually using the theory of quasimodular forms and quasimock modular forms. As an application, we find asymptotic series for the moments of the partition rank and crank statistics. Although the coefficients are exponentially large, the error in the series expansions is polynomial, and have the same order as the coefficients of the residual Eisenstein series that are undetectable by the Circle Method. We also prove asymptotic series expansions for the symmetrized rank and crank moments introduced by Andrews and Garvan, respectively. Equivalently, the former gives asymptotic series for the enumeration of Andrews k-marked Durfee symbols.