Recently Gordon and McIntosh introduced the third order mock theta function $ξ(q)$ defined by $$ ξ(q)=1+2\sum_{n=1}^{\infty}\frac{q^{6n^2-6n+1}}{(q;q^6)_{n}(q^5;q^6)_{n}}. $$ Our goal in this paper is to study arithmetic properties of the coefficients of this function. We present a number of such properties, including several infinite families of Ramanujan--like congruences.
We study the divisibility properties of the partition function associated with the eighth order mock theta function $V_0(q)$, introduced by Gordon and McIntosh. We obtain congruences modulo powers of 2 for certain coefficients of the partition function, akin to Ramanujan's partition congruences. Further, we also present several infinite families of congruences molulo 13, 25 and 27.
We exhibit an infinite family of vector-valued mock theta functions indexed by positive integers coprime to $6$. These are built from specializations of Dyson's rank generating function and related functions studied by Watson, Gordon, and McIntosh. The associated completed harmonic Maass forms transform according to the Weil representation attached to a rank one lattice. This strengthens a 2010 result of Bringmann and Ono and a 2019 result of Garvan.
In recent work of Bringmann, Guerzhoy, and the first author, p-adic modular forms were constructed from mock modular forms. This paper proves explicit congruences for these p-adic modular forms.
Let $β(q)=\sum_{n\ge 0} \mathfrak{b}(n)q^n$ be a second order mock theta function defined by $$\sum_{n\ge 0}\frac{q^{n(n+1)}(-q^2;q^2)_n}{(q;q^2)_{n+1}^2}.$$ In this paper, we obtain an infinite family of congruences modulo powers of $3$ for $\mathfrak{b}(n)$.
Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third order mock theta functions $ω(q)$ and $ν(q)$. In this paper, we find several new exact generating functions for those partition functions as well as the associated smallest parts functions and deduce several new congruences modulo powers of 5.
In 1982, Gessel showed that the Apéry numbers associated to the irrationality of $ζ(3)$ satisfy Lucas congruences. Our main result is to prove corresponding congruences for all sporadic Apéry-like sequences. In several cases, we are able to employ approaches due to McIntosh, Samol--van Straten and Rowland--Yassawi to establish these congruences. However, for the sequences often labeled $s_{18}$ and $(η)$ we require a finer analysis. As an application, we investigate modulo which numbers these sequences are periodic. In particular, we show that the Almkvist--Zudilin numbers are periodic modulo …
In a private communication, K. Ono conjectured that any mock theta function of weight 1/2 or 3/2 can be congruent modulo a prime $p$ to a weakly holomorphic modular form for just a few values of $p$. In this paper we describe when such a congruence occurs. More precisely we show that it depends on the $p$-adic valuation of the mock theta function itself. In order to do so, we construct a family of mock theta functions in terms of derivatives of the Appell sum, which have a special Fourier expansion at infinity.
Using a family of mock modular forms constructed by Zagier, we study the coefficients of a mock modular form of weight $3/2$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo primes $\ell\geq 5$. These coefficients are related to the smallest parts function of Andrews. As an application, we reprove a theorem of Garvan regarding the properties of this function modulo $\ell$. As another application, we show that congruences modulo $\ell$ for the smallest parts function are rare in a precise sense.
In 2012 Bryson, Ono, Pitman and Rhoades showed how the generating functions for certain strongly unimodal sequences are related to quantum modular and mock modular forms. They proved some parity results and conjectured some mod 4 congruences for the coefficients of these generating functions. In 2016 Kim, Lim and Lovejoy obtained similar results for odd-balanced unimodal sequences and made similar mod 4 conjectures. We prove all of these mod 4 conjectures and similar congruences for the Andrews spt-function and related mock theta functions. Our method of proof involves new Hecke-Rogers type id…
In Ramanujan's final letter to Hardy, he listed examples of a strange new class of infinite series he called "mock theta functions". It turns out all of these examples are essentially specializations of a so-called universal mock theta function $g_3(z,q)$ of Gordon-McIntosh. Here we show that $g_3$ arises naturally from the reciprocal of the classical Jacobi triple product -- and is intimately tied to rank generating functions for unimodal sequences, which are connected to mock modular and quantum modular forms -- under the action of an operator related to statistical physics and partition the…
It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for rank parity function is f(q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.
In this paper, we discuss a few recent conjectures made by George Beck related to the ranks and cranks of partitions. The conjectures for the rank of a partition were proved by Andrews by using results due to Atkin and Swinnerton-Dyer on a suitable generating function, while the conjectures related to cranks were studied by Shane Chern using weighted partition moments. We revisit the conjectures on the crank of a partition by decomposing the relevant generating function and further explore connections with Apple-Lerch series and tenth order mock theta functions.
We survey divisibility properties of the Fourier coefficients of modular functions inspired by Ramanujan. Then using recent results of the generalized Hecke operator on harmonic Maass functions and known divisibility of Fourier coefficients of modular functions, we establish congruence relations of the Fourier coefficients of certain modular functions and mock modular functions of various levles.
It was recently shown that $qω(q)$, where $ω(q)$ is one of the third order mock theta functions, is the generating function of $p_ω(n)$, the number of partitions of a positive integer $n$ such that all odd parts are less than twice the smallest part. In this paper, we study the overpartition analogue of $p_ω(n)$, and express its generating function in terms of a ${}_3ϕ_{2}$ basic hypergeometric series and an infinite series involving little $q$-Jacobi polynomials. This is accomplished by obtaining a new seven parameter $q$-series identity which generalizes a deep identity due to the first auth…
We establish two families of congruences modulo powers of 5 for the Fourier coefficients of $(2E_2(2τ)-E_2(τ))η(2τ)^{-1}$, where $E_2(τ)$ is the weight 2 Eisenstein series and $η(τ)$ is the Dedekind eta function. This allows us to prove similar congruences for two smallest parts functions. The first function is $\mathrm{spt}_ω(n)$, which was introduced by Andrews, Dixit and Yee and associated with Ramanujan/Watson's third order mock theta function $ω(q)$. The second one is $\mathrm{spt}_{C5}(n)$, which appeared in the work of Garvan and Jennings-Shaffer. Moreover, we confirm two conjectural co…
The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function $ω(q)$ (resp. $ν(-q)$). Similar results for partitions with the corresponding restriction on each even part are also obtained, one of which involves the third order mock theta function $ϕ(q)$. Congruences for the smallest parts functions associated to such partitions are obtained. Two analogues of the partition-theoretic interpretation of Euler's pentagonal theorem are also obtained.
At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of $R(ζ,q)$, where $R(z,q)$ is the two-variable generating function of Dyson's rank function and $ζ$ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon and McIntosh, and motivated by Dyson's question, Bringmann, Ono and Rhoades studied transformation properties of $R(ζ,q)$. In this paper we strengthen and extend the results of Bringma…
In a recent paper, Andrews, Dixit, and Yee introduced a new spt-type function $\operatorname{spt}_ω(n)$, which is closely related to Ramanujan's third order mock theta function $ω(q)$. Garvan and Jennings-Shaffer introduce a crank function which explains congruences for $\operatorname{spt}_ω(n)$. In this note, we study asymptotic behavior of this crank function and confirm a positivity conjecture of the crank asymptotically. We also study a sign pattern of the crank and congruences for $\operatorname{spt}_ω(n)$.
On page 3 of his lost notebook, Ramanujan defines the Appell-Lerch sum $$ϕ(q):=\sum_{n=0}^\infty \dfrac{(-q;q)_{2n}q^{n+1}}{(q;q^2)_{n+1}^2},$$ which is connected to some of his sixth order mock theta functions. Let $\sum_{n=1}^\infty a(n)q^n:=ϕ(q)$. In this paper, we find a representation of the generating function of $a(10n+9)$ in terms of $q$-products. As corollaries, we deduce the congruences $a(50n+19)\equiv a(50n+39)\equiv a(50n+49)\equiv0~(\textup{mod}~25)$ as well as $a(1250n+250r+219)\equiv 0~(\textup{mod}~125)$, where $r=1$, $3$, and $4$. The first three congruences were conjectured …
By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) "small divisors" summatory functions $σ_ψ^{\mathrm{sm}}(n)$. More precisely, in terms of the weight 2 quasimodular Eisenstein series $E_2(τ)$ and a generic Shimura theta function $θ_ψ(τ)$, we show that there is a constant $α_ψ$ for which $$ \math…
The study of arithmetic properties of coefficients of modular forms $f(τ) = \sum a(n)q^n$ has a rich history, including deep results regarding congruences in arithmetic progressions. Recently, work of C.-S. Radu, S. Ahlgren, B. Kim, N. Andersen, and S. Löbrich have employed the $q$-expansion principle of P. Deligne and M. Rapoport in order to determine more about where these congruences can occur. Here, we extend the method to give additional results for a large class of modular forms. We also give analogous results for generalized Frobenius partitions and the two mock theta functions $f(q)$ a…
In 2018 Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third order mock theta function $ω(q)$. Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 $\mathbb{Z}[X]$-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang's family of congruences, which is associated with a genus 0 modular curve, can …
