We develop a generalized version of the Hardy-Ramanujan "circle method" in order to derive asymptotic series expansions for the products of modular forms and mock theta functions. Classical asymptotic methods (including the circle method) do not work in this situation because such products are not modular, and in fact, the "error integrals" that occur in the transformations of the mock theta functions can (and often do) make a significant contribution to the asymptotic series. The resulting series include principal part integrals of Bessel functions, whereby the main asymptotic term can also be identified. To illustrate the application of our method, we calculate the asymptotic series expansion for the number of partitions without sequences. Andrews showed that the generating function for such partitions is the product of the third order mock theta function $\chi$ and a (modular) infinite product series. The resulting asymptotic expansion for this example is particularly interesting because the error integrals in the modular transformation of the mock theta function component appear in the exponential main term.

Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dyson's rank statistic. He used these connections to find identities relating their generating functions as well as to prove Ramanujan-type congruences for these objects and find relations between.In this paper we show that the hypergeometric generating functions for these objects are natural examples of quasimock theta functions, which are defined as the holomorphic parts of weak Maass forms and their derivatives. In particular, these generating functions may be viewed as analogs of Ramanujan's mock theta functions with arbitrarily high weight. We use the automorphic properties to prove the existence of infinitely many congruences for the Durfee symbols. Furthermore, we show that as k varies, the modularity of the k-marked Durfee symbols is precisely dictated by the case k = 2. Finally, we use this relation in order to prove the existence of general congruences for rank moments in terms of level one modular forms of bounded weight.

In 1944, Freeman Dyson conjectured the existence of a ''crank'' function for partitions that would provide a combinatorial proof of Ramanujan's congruence modulo 11. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that the crank simultaneously ''explains'' the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.

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.

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, and our main result states that while the two families of moment functions are asymptotically equal, the crank moments are always asymptotically larger than the rank moments.Andrews, Chan, and Kim primarily focused on one case, and proved the stronger result that the first crank moment is strictly larger than the first rank moment for all partitions by showing that the difference is equal to a combinatorial statistic on partitions that they named the ospt-function. Our main results therefore also give the asymptotic behavior of the ospt-function, and we further determine its behavior modulo 2 by relating its parity to Andrews spt-function.