A note on the equidistribution of 33-colour partitions

In this short note, we prove equidistribution results regarding three families of three-colour partitions recently introduced by Schlosser and Zhou. To do so, we prove an asymptotic for the infinite product $F_{a,c}(\zeta ; q) \coloneqq \prod_{n \geq 0} \left(1- \zeta q^{a+cn}\right)$ ($a,c \in \N$ with $0<a\leq c$ and $\zeta$ a root of unity) in certain cones in the right half-plane, which may be useful in studying similar problems. As a corollary, we obtain the asymptotic behaviour of the three-colour partition families at hand.Comment: Short 7-page note detailing some asymptotics for functions introduced by Schlosser & Zhou last wee

A short note on higher Mordell integrals

In this short note we show that techniques of Bringmann, Kaszian, and Milas hold for computing the higher Mordell integrals associated to a general family of quantum modular forms of depth two and weight one.Comment: 8 page

Modular Forms: Constructions & Applications

This thesis combines results of five research papers on the construction and applications of modular forms and their generalisations. We begin by constructing new examples of quantum modular forms of depth two, generalising results of Bringmann, Kaszian, and Milas. To do so, we relate the asymptotics of certain false theta functions of binary quadratic forms to multiple Eichler integrals of theta functions. Quantum modularity of the false theta functions follows from the behaviour of such integrals near the real line. Next, we turn our attention to the asymptotic profile of a certain eta-theta quotient that arises in the partition function of entanglement entropy in string theory. In particular, we generalise methods of Bringmann and Dousse, and Dousse and Mertens, to deal with the meromorphic Jacobi form at hand. Applying Wright's circle method for Jacobi forms we obtain a bivariate asymptotic for the two-variable coefficients of the eta-theta quotient. Thirdly, we investigate the asymptotic behaviour of the generating function of integer partitions whose ranks are congruent to r modulo t, denoted by N(r; t; n). By proving that the series has monotonic increasing coefficients above some bound, we are in a position to apply Ingham's Tauberian theorem. This immediately implies that N(r; t; n) is equidistributed in r for fixed t as n tends to infinity, in turn implying a recent conjecture of Hou and Jagadeeson on a convexity-type result. The following chapter is dedicated to an investigation of traces of cycle integrals of meromorphic modular forms and their relationship to coefficients of harmonic Maass forms. Working on lattices of signature (1,2), we first relate a locally harmonic Maass form to a Siegel theta lift involving the Maass raising operator by explicitly computing the raising of the locally harmonic Maass form, and using the usual unfolding argument for the theta lift. We then borrow techniques of Bruinier, Ehlen, and Yang to compute the theta lift as (up to terms that vanish for certain classes of input functions) the constant term in a q-series involving the coefficients of xi-preimages of unary theta functions as well as theta functions. Since such preimages are harmonic Maass forms, we obtain a description of the traces in terms of coefficients of theta functions and harmonic Maass forms. Choosing a specific lattice related to quadratic forms and noting that the functions determining the constant term can be chosen to have rational coefficients, we obtain a new proof of a recent result of Alfes-Neumann, Bringmann, and Schwagenscheidt. Finally, we investigate the relationship between modular forms and self-conjugate t-core partitions. We obtain the number of self-conjugate 7-cores as a single class number in two ways. The first we show with modularity arguments on the generating function of Hurwitz class numbers. We also provide a complementary combinatorial description to explain the equality. In particular, we construct an explicit map between self-conjugate t-cores and quadratic forms in a given class group. Moreover, we show that the genus of the quadratic forms is unique, and determine the number of preimages of the genus. Using these results, we show an equality between the number of 4-cores and the number of self-conjugate 7-cores on specific arithmetic progressions. Aside from the t = 4 case, we consider whether equalities between t-cores and self-conjugate 2t-1-cores are possible. We show for t = 2,3,5 that they are not, and offer a conjecture and partial results for t > 5

On tt-core and self-conjugate (2t−1)(2t-1)-core partitions in arithmetic progressions

We extend recent results of Ono and Raji, relating the number of self-conjugate $7$-core partitions to Hurwitz class numbers. Furthermore, we give a combinatorial explanation for the curious equality $2\operatorname{sc}_7(8n+1) = \operatorname{c}_4(7n+2)$. We also conjecture that an equality of this shape holds if and only if $t=4$, proving the cases $t\in\{2,3,5\}$ and giving partial results for $t>5$

Local Maa{\ss} forms and Eichler--Selberg type relations for negative weight vector-valued mock modular forms

By comparing two different evaluations of a modified (\`{a} la Borcherds) higher Siegel theta lift on even lattices of signature $(r,s)$, we prove Eichler--Selberg type relations for a wide class of negative weight vector-valued mock modular forms. In doing so, we detail several properties of the lift, as well as showing that it produces an infinite family of local (and locally harmonic) Maa{\ss} forms on Grassmanians in certain signatures.Comment: 21 pages, no figures, comments welcom

Asymptotics for dd-fold partition diamonds and related infinite products

We prove an asymptotic formula for the number of $d$-fold partition diamonds of $n$ and their Schmidt-type counterparts. In order to do so, we study the asymptotic behavior of certain infinite products. We also remark on interesting potential connections with mathematical physics and Bloch groups.Comment: 20 pages, comments welcome. Second draft corrects some remark

Equidistribution and partition polynomials

Using equidistribution criteria, we establish divisibility by cyclotomic polynomials of several partition polynomials of interest, including $spt$-crank, overpartition pairs, and $t$-core partitions. As corollaries, we obtain new proofs of various Ramanujan-type congruences for associated partition functions. Moreover, using results of Erd\"os and Tur\'an, we establish the equidistribution of roots of partition polynomials on the unit circle including those for the rank, crank, $spt$, and unimodal sequences. Our results complement earlier work on this topic by Stanley, Boyer-Goh, and others. We explain how our methods may be used to establish similar results for other partition polynomials of interest, and offer many related open questions and examples.Comment: 15 pages, 8 figures. This iteration fixes several typos, adds some discussion, and asks slightly more precise question