arXiv subjects

Joshua Males

Publications and source records attributed to Joshua Males.

At least 19 recordsLinked to original sources

arXiv2023

A note on hook length equidistribution on arithmetic progressions

In a recent paper, Bringmann, Craig, Ono, and the author showed that the number of $t$-hooks ($t\geq2$) among all partitions of $n$ is not always asymptotically equidistributed on congruence classes $a \pmod{b}$. In this short note, we clarify the situation of $t=1$, i.e. all hook lengths, and show that this case does give asymptotic equidistribution, closing the story of the distribution properties of $t$-hooks on congruence classes.

math.CO↗

arXiv2023

Asymptotics for $d$-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.

math.NT↗

arXiv2023

Crank equidistribution and $(k,j)$-overlined partitions

In a paper published in 2023, Wagner introduced and studied Jacobi forms with complex multiplication, and gave several applications. One such application was in constructing a new doubly-infinite family of partition-theoretic objects, called $(k,j)$-coloured overpartitions and labelled by $\overline{p}_{k,j}$, and using the Jacobi forms to construct crank functions which explain the Ramanujan-type congruences satisfied by $\overline{p}_{k,j}$. In this note, we give an asymptotic formula for the number of $(k,j)$-coloured overpartitions and prove that any crank constructed by Wagner is asymptotically equidistributed on arithmetic progressions, following several recent papers in the literature.

math.NT↗

arXiv2023

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 Erdös and Turán, 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.

math.NT↗

arXiv2023

The second shifted difference of partitions and its applications

A number of recent papers have estimated ratios of the partition function $p(n-j)/p(n)$, which appears in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study second shifted difference of partitions, $f(j,n):= p(n) -2p(n-j) +p(n-2j)$, and give another easy-to-use estimate of $f(j,n)$. As applications of these, we prove a shifted convexity property of $p(n)$, as well as giving new estimates of the $k$-rank partition function $N_k(m,n)$ and non-$k$-ary partitions along with their differences.

math.NT↗

arXiv2023

A note on the equidistribution of $3$-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 formula for the infinite product $F_{a,c}(\zeta ; {\rm e}^{-z}) := \prod_{n \geq 0} \big(1- \zeta {\rm e}^{-(a+cn)z}\big)$ ($a,c \in \mathbb{N}$ with $0<a\leq c$ and $\zeta$ a root of unity) when $z$ lies in certain sectors 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.

math.CO↗

arXiv2023

Jensen polynomials associated with Wright's circle method: Hyperbolicity and Turán inequalities

We study the Fourier coefficients of functions satisfying a certain version of Wright's circle method with finitely many major arcs. We show that the Jensen polynomials associated with such Fourier coefficients are asymptotically hyperbolic, building on the framework of Griffin--Ono--Rolen--Zagier and others. Consequently, we prove that the Fourier coefficients asymptotically satisfy all higher-order Turán inequalities. As an application, we apply our results to both $(q^t;q^t)_\infty^{-r}$, which counts $r$-coloured partitions into parts divisible by $t$, and to the function $(q^{a};q^{p})_\infty^{-1}$ where $p$ is prime and $0\leq a<p$, a ubiquitous function throughout number theory.

math.NT↗

arXiv2023

Central $L$-values of newforms and local polynomials

In this paper, we characterize the vanishing of twisted central $L$-values attached to newforms of square-free level in terms of certain polynomials of quadratic forms introduced by Zagier and the action of finitely many Hecke operators thereon. To be more precise, we establish that a twisted central $L$-value attached to a newform vanishes if and only if a certain explicitly computable polynomial is constant. We describe these constants explicitly in two different ways. One of the descriptions involves the generalized Hurwitz class numbers, which were introduced by Pei and Wang in $2003$. We provide some numerical examples and conclude by offering some questions for future work.

math.NT↗

arXiv2023

Oscillating asymptotics and conjectures of Andrews

In 1986, Andrews studied the function $\sigma(q)$ from Ramanujan's ``Lost" Notebook, and made several conjectures on its Fourier coefficients $S(n)$, which count certain partition ranks. In 1988, Andrews-Dyson-Hickerson famously resolved these conjectures, relating the coefficients $S(n)$ to the arithmetic of $\mathbb Q(\sqrt{6})$; this relationship was further expounded upon by Cohen in his work on Maass waveforms, and was more recently extended by Zwegers and by Li and Roehrig. A closer inspection of Andrews' original work on $\sigma(q)$ reveals additional related functions and conjectures, which we study in this paper. In particular, we study the function $v_1(q)$, also from Ramanujan's ``Lost" Notebook, a $q$-hypergeometric series with partition-theoretic Fourier coefficients $V_1(n)$, and prove two of Andrews' conjectures on $V_1(n)$ which are parallel to his original conjectures on $S(n)$. Our methods differ from those used by Andrews-Dyson-Hickerson, and require a blend of novel techniques inspired by Garoufalidis' and Zagier's recent work on asymptotics of Nahm sums, with classical techniques including the Circle Method in Analytic Number Theory; our methods may also be applied to determine the asymptotic behavior of similar $q$-hypergeometric series of interest which are not amenable to classical techniques. We also offer explanations of additional related conjectures of Andrews, ultimately connecting the asymptotics of $V_1(n)$ to the arithmetic of $\mathbb Q(\sqrt{-3})$.

math.NT↗

arXiv2022

Sign changes in statistics for plane partitions

Recent work of Cesana, Craig and the third author shows that the trace of plane partitions is asymptotically equidistributed in residue classes mod $b$. Applying a technique of the first two authors and Garnowski, we prove asymptotic formulas for the secondary terms in this equidistribution, which are controlled by certain complex numbers generated by a twisted MacMahon-type product. We further carry out a similar analysis for a statistic related to plane overpartitions.

math.NT↗

arXiv2022

Asymptotics, Turán inequalities, and the distribution of the BG-rank and 2-quotient rank of partitions

Let $j,n$ be even positive integers, and let $\overline{p}_j(n)$ denote the number of partitions with BG-rank $j$, and $\overline{p}_j(a,b;n)$ to be the number of partitions with BG-rank $j$ and $2$-quotient rank congruent to $a \pmod{b}$. We give asymptotics for both statistics, and show that $\overline{p}_j(a,b;n)$ is asymptotically equidistributed over the congruence classes modulo $b$. We also show that each of $\overline{p}_j(n)$ and $\overline{p}_j(a,b;n)$ asymptotically satisfy all higher-order Turán inequalities.

math.NT↗

arXiv2022

Combinatorial results on $t$-cores and sums of squares

We classify the connection between $t$-cores and self-conjugate $t$-cores to sums of squares. To do so, we provide explicit maps between $t$-core partitions and self-conjugate $t$-core partitions of a positive integer $n$ to representations of certain numbers as sums of squares. For example, the self-conjugate $4$-core partition $λ=(4,1,1,1)$ corresponds uniquely to the solution $61=6^2+5^2$. As a corollary, we completely classify the relationship between $t$-cores and Hurwitz class numbers. Using these tools, we see how certain sets of representations as sums of squares naturally decompose into families of $t$-cores. Finally, we construct an explicit map on partitions to explain the equality $2\operatorname{sc}_7(8n+1) = \operatorname{c}_4(7n+2)$ previously studied by Bringmann, Kane, and the first author.

math.CO↗

arXiv2022

Higher Siegel theta lifts on Lorentzian lattices, harmonic Maass forms, and Eichler-Selberg type relations

We investigate so-called "higher" Siegel theta lifts on Lorentzian lattices in the spirit of Bruinier-Ehlen-Yang and Bruinier-Schwagenscheidt. We give a series representation of the lift in terms of Gauss hypergeometric functions, and evaluate the lift as the constant term of a Fourier series involving the Rankin-Cohen bracket of harmonic Maass forms and theta functions. Using the higher Siegel lifts, we obtain a vector-valued analogue of Mertens' result stating that the Rankin--Cohen bracket of the holomorphic part of a harmonic Maass form of weight $\frac{3}{2}$ and a unary theta function, plus a certain form, is a holomorphic modular form. As an application of these results, we offer a novel proof of a conjecture of Cohen which was originally proved by Mertens, as well as a novel proof of a theorem of Ahlgren and Kim, each in the scalar-valued case.

math.NT↗

arXiv2022

Distributions on partitions arising from Hilbert schemes and hook lengths

Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory, and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations which arise from extensions of the Nekrasov-Okounkov hook product formula, and from Betti numbers of various Hilbert schemes of $n$ points on $\mathbb{C}^2.$ For the Hilbert schemes, we prove that homology is equidistributed as $n\to \infty.$ For $t$-hooks, we prove distributions which are often not equidistributed. The cases where $t\in \{2, 3\}$ stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results which are of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products $$ F_1(ξ; q):=\prod_{n=1}^{\infty}\left(1-ξq^n\right), \ \ \ F_2(ξ; q):=\prod_{n=1}^{\infty}\left(1-(ξq)^n\right) \ \ \ {\text {and}}\ \ \ F_3(ξ; q):=\prod_{n=1}^{\infty}\left(1-ξ^{-1}(ξq)^n\right). $$

math.NT↗

arXiv2022

Exact formulae and Turán inequalities for Vafa-Witten invariants of $K3$ surfaces

We consider three different families of Vafa-Witten invariants of $K3$ surfaces. In each case, the partition function that encodes the Vafa-Witten invariants is given by combinations of twisted Dedekind $η$-functions. By utilising known properties of these $η$-functions, we obtain exact formulae for each of the invariants and prove that they asymptotically satisfy all higher-order Turán inequalities.

math.NT↗

arXiv2021

A short note on higher Mordell integrals

Classical mock modular and quantum modular forms are known to have an intimate relationship with Mordell integrals thanks to Zwegers' groundbreaking PhD thesis. More recently, generalisations of mock/quantum modular forms to so-called "higher depth" versions have been intensively studied. In essence, a mock/quantum modular form of depth $d$ is such that the error of modularity transforms as another mock/quantum modular form of depth $d - 1$. In this short note we use techniques of Bringmann, Kaszian, and Milas to show that the double Eichler integrals of a family of depth two quantum modular forms of weight one previously studied by the author can be related to certain "higher" Mordell integrals, meaning it may be written as a certain double integral, a la Zwegers.

math.NT↗

arXiv2021

Asymptotic equidistribution for partition statistics and topological invariants

We provide a general framework for proving asymptotic equidistribution, convexity, and log concavity of coefficients of generating functions on arithmetic progressions. Our central tool is a variant of Wright's Circle Method proved by two of the authors with Bringmann and Ono, following work of Ngo and Rhoades. We offer a selection of different examples of such results, proving asymptotic equidistribution results for several partition statistics, modular sums of Betti numbers of two- and three-flag Hilbert schemes, and the number of cells of dimension a (mod b) of a certain scheme central in work of Göttsche.

math.NT↗

arXiv2021

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.

math.NT↗