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 be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of $\mathbb{Z}[X]$. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic growth. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.
In 2017, Andrews, Dixit, Schultz and Yee introduced the function $\overline{\textrm{spt}}_ω(n)$, which denotes the number of smallest parts in the overpartitions of $n$ in which the smallest part is always overlined and all odd parts are less than twice the smallest part. Recently, Baruah and Begum established several internal congruences and congruences modulo small powers of $5$ for $\overline{\textrm{spt}}_ω(n)$. Moreover, they conjectured a family of internal congruences modulo any powers of $5$ and two families of congruences modulo any even powers of $5$. In this paper, we confirm three …
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…
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…
Atkin and Garvan introduced the functions $N_k(n)$ and $M_k(n)$, which denote the $k$-th moments of ranks and cranks in the theory of partitions. Let $e_{2r}(n)$ be the $n$-th Fourier coefficient of $E_{2r}(τ)/η(τ)$, where $E_{2r}(τ)$ is the classical Eisenstein series of weight $2r$ and $η(τ)$ is the Dedekind eta function. Via the theory of quasi-modular forms, we find that for $k \leq 5$, $N_k(n)$ and $M_k(n)$ can be expressed using $e_{2r}(n)$ ($2\leq r \leq k$), $p(n)$ and $N_2(n)$. For $k>5$, additional functions are required for such expressions. For $r\in \{2, 3, 4, 5, 7\}$, by studying…
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…
