Divisibility of Andrews' Singular Overpartitions by Powers of 2 and 3

TL;DR

This paper proves that the singular overpartition function introduced by Andrews is almost always divisible by arbitrarily high powers of 2 and 3, extending previous divisibility results.

Contribution

It establishes that for any positive integer k, the function C_{3,1}(n) is almost always divisible by 2^k and 3^k, generalizing earlier divisibility findings.

Findings

C_{3,1}(n) is almost always divisible by 2^k for any k.

C_{3,1}(n) is almost always divisible by 3^k for any k.

Extends previous divisibility results for singular overpartitions.

Abstract

Andrews introduced the partition function $\overline{C}_{k, i}(n)$, called singular overpartition, which counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i\pmod{k}$ may be overlined. He also proved that $\overline{C}_{3, 1}(9n+3)$ and $\overline{C}_{3, 1}(9n+6)$ are divisible by $3$ for $n\geq 0$. Recently Aricheta proved that for an infinite family of $k$, $\overline{C}_{3k, k}(n)$ is almost always even. In this paper, we prove that for any positive integer $k$, $\overline{C}_{3, 1}(n)$ is almost always divisible by $2^k$ and $3^k.$

Equations

$$\sum _{n=0}^{\infty }{\overline{C}}_{k,i}(n)q^n=\frac{(q^k;q^k{)}_{\infty }(-q^i;q^k{)}_{\infty }(-q^{k-i};q^k{)}_{\infty }}{(q;q{)}_{\infty }},$$

$${\overline{C}}_{3,1}(9n+3)\equiv {\overline{C}}_{3,1}(9n+6)\equiv 0(\mathrm{mod}3).$$

$$\underset{X\to \infty }{\lim }\frac{\#\left\{0<n\le X:{\overline{C}}_{3k,k}(n)\equiv 0(\mathrm{mod}2)\right\}}{X}$$

$$\underset{X\to \infty }{\lim }\frac{\#\left\{0<n\le X:{\overline{C}}_{3,1}(n)\equiv 0(\mathrm{mod}{2}^k)\right\}}{X}$$

$$\underset{X\to \infty }{\lim }\frac{\#\left\{0<n\le X:{\overline{C}}_{3,1}(n)\equiv 0(\mathrm{mod}{3}^k)\right\}}{X}$$

$${\text{SL}}_2(\mathbb{Z})$$

$${\Gamma }_0(N)$$

$${\Gamma }_1(N)$$

$$\Gamma (N)$$

$${\text{GL}}_2^{+}(\mathbb{R})=\left\{\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]:a,b,c,d\in \mathbb{R}\text{and}ad-bc>0\right\}$$

$$(f{|}_{\ell }\gamma )(z):=(\text{det}\gamma {)}^{\ell /2}(cz+d{)}^{-\ell }f(\gamma z).$$

$$f\left(\frac{az+b}{cz+d}\right)=(cz+d{)}^{\ell }f(z)$$

$$(f{|}_{\ell }\gamma )(z)=\sum _{n\ge 0}{a}_{\gamma }(n)q_N^n,$$

$$f\left(\frac{az+b}{cz+d}\right)=\chi (d)(cz+d{)}^{\ell }f(z)$$

$$\eta (z):=q^{1/24}(q;q{)}_{\infty }=q^{1/24}\prod _{n=1}^{\infty }(1-q^n),$$

$$f(z)=\prod _{\delta \mid N}\eta (\delta z{)}^{r_{\delta }},$$

$$\sum _{\delta \mid N}\delta r_{\delta }\equiv 0(\mathrm{mod}24)$$

$$\sum _{\delta \mid N}\frac{N}{\delta }{r}_{\delta }\equiv 0(\mathrm{mod}24),$$

$$f\left(\frac{az+b}{cz+d}\right)=\chi (d)(cz+d{)}^{\ell }f(z)$$

$$\frac{N}{24}\sum _{\delta \mid N}\frac{\gcd (d,\delta {)}^2{r}_{\delta }}{\gcd (d,\frac{N}{d})d\delta }.$$

$$\sum _{n=0}^{\infty }{\overline{C}}_{3,1}(n)q^n=\frac{(q^2;q^2{)}_{\infty }(q^3;q^3{)}_{\infty }^2}{(q;q{)}_{\infty }^2(q^6;q^6{)}_{\infty }}.$$

$$A_p(z)=\prod _{n=1}^{\infty }\frac{(1-q^{24n}{)}^p}{(1-q^{24pn})}=\frac{{\eta }^p(24z)}{\eta (24pz)}.$$

$$A_p^{p^k}(z)=\frac{{\eta }^{p^{k+1}}(24z)}{{\eta }^{p^k}(24pz)}\equiv 1(\mathrm{mod}{p}^{k+1}).$$

$$B_{p,k}(z)=\left(\frac{\eta (48z)\eta (72z{)}^2}{\eta (24z{)}^2\eta (144z)}\right)A_p^{p^k}(z).$$

$$B_{p,k}(z)\equiv \frac{\eta (48z)\eta (72z{)}^2}{\eta (24z{)}^2\eta (144z)}=\frac{(q^{48};q^{48}{)}_{\infty }(q^{72};q^{72}{)}_{\infty }^2}{(q^{24};q^{24}{)}_{\infty }^2(q^{144};q^{144}{)}_{\infty }}.$$

$$B_{p,k}(z)\equiv \sum _{n=0}^{\infty }{\overline{C}}_{3,1}(n)q^{24n}(\mathrm{mod}{p}^{k+1}).$$

$$B_{2,k}(z)=\left(\frac{\eta (48z)\eta (72z{)}^2}{\eta (24z{)}^2\eta (144z)}\right)A_2^{2^k}(z)=\frac{\eta (24z{)}^{2^{k+1}-2}\eta (72z{)}^2}{\eta (48z{)}^{2^k-1}\eta (144z)}.$$

$$\frac{\gcd (d,24{)}^2}{24}\left(2^{k+1}-2\right)+\frac{\gcd (d,48{)}^2}{48}\left(1-2^k\right)+\frac{\gcd (d,72{)}^2}{36}-\frac{\gcd (d,144{)}^2}{144}\ge 0.$$

$$D:=6\frac{\gcd (d,24{)}^2}{\gcd (d,144{)}^2}\left(2^{k+1}-2\right)+3\frac{\gcd (d,48{)}^2}{\gcd (d,144{)}^2}\left(1-2^k\right)+4\frac{\gcd (d,72{)}^2}{\gcd (d,144{)}^2}-1\ge 0.$$

$$f(z)=\sum _{n=0}^{\infty }c(n)q^n\in \mathbb{Z}[[q]],$$

$$\#\left\{n\le X:c(n)\not\equiv 0(\mathrm{mod}m)\right\}=\mathcal{O}\left(\frac{X}{(\log X{)}^{\alpha }}\right).$$

Full text

1. Introduction and statement of results

In [7], Corteel and Lovejoy introduced overpartitions. An overpartition of $n$ is a non-increasing sequence of natural numbers whose sum is $n$ in which the first occurrence of a number may be overlined. In [2], Andrews defined the partition function $\overline{C}_{k,i}(n)$, called singular overpartition, which counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv\pm i\pmod{k}$ may be overlined. For example, $\overline{C}_{3,1}(4)=10$ with the relevant partitions being $4,\overline{4},2+2,\overline{2}+2,2+1+1,\overline{2}+1+1,2+\overline{1}+1,\overline{2}+\overline{1}+1,1+1+1+1,\overline{1}+1+1+1$. For $k\geq 3$ and $1\leq i\leq\left\lfloor\frac{k}{2}\right\rfloor$, the generating function for $\overline{C}_{k,i}(n)$ is given by

[TABLE]

where $\displaystyle(a;q)_{\infty}:=\prod_{j=0}^{\infty}(1-aq^{j})$. Andrews proved the following Ramanujan-type congruences satisfied by $\overline{C}_{3,1}(n)$: For $n\geq 0$,

[TABLE]

Chen, Hirschhorn and Sellers [6] later showed that Andrews’ congruences modulo $3$ are two examples of an infinite family of congruences modulo $3$ which hold for the function $\overline{C}_{3,1}(n)$. Recently, Ahmed and Baruah [1] found congruences modulo $4$, $18$ and $36$ for $\overline{C}_{3,1}(n)$, infinite families of congruences modulo $2$ and $4$ for $\overline{C}_{8,2}(n)$, congruences modulo $2$ and $3$ for $\overline{C}_{12,2}(n)$ and $\overline{C}_{12,4}(n)$; and congruences modulo $2$ for $\overline{C}_{24,8}(n)$ and $\overline{C}_{48,16}(n)$.

In a recent work, Naika and Gireesh [11] proved congruences for $\overline{C}_{3,1}(n)$ modulo $6$, $12$, $16$, $18$, and $24$. They also found infinite families of congruences for $\overline{C}_{3,1}(n)$ modulo $12$, $18$, $48$, and $72$. In [4], we affirm a conjecture of Naika and Gireesh by proving that $\overline{C}_{3,1}(12n+11)\equiv 0\pmod{144}$ for all $n\geq 0$.

In [6], Chen, Hirschhorn and Sellers studied the parity of $\overline{C}_{k,i}(n)$. They showed that $\overline{C}_{3,1}(n)$ is always even and that $\overline{C}_{6,2}(n)$ is even (or odd) if and only if $n$ is not (or is) a pentagonal number. In a very recent paper, Aricheta [3] studied the parity of $\overline{C}_{3k,k}(n)$. To be specific, represent any positive integer $k$ as $k=2^{a}m$ where the integer $a\geq 0$ and $m$ is positive odd. Assume further that $2^{a}\geq m$. Then Aricheta proved that $\overline{C}_{3k,k}(n)$ is almost always even, that is

[TABLE]

Let $k$ be a fixed positive integer. Gordon and Ono [8] proved that the number of partitions of $n$ into distinct parts is divisible by $2^{k}$ for almost all $n$. Similar studies are done for some other partition functions, for example see [5, 10, 14]. In this article, we study divisibility of $\overline{C}_{3,1}(n)$ by $2^{k}$ and $3^{k}$. To be specific, we prove that $\overline{C}_{3,1}(n)$ is divisible by $2^{k}$ and $3^{k}$ for almost all $n$.

Theorem 1.1**.**

Let $k$ be a fixed positive integer. Then $\overline{C}_{3,1}(n)$ is almost always divisible by $2^{k}$, namely,

[TABLE]

We further prove that the partition function $\overline{C}_{3,1}(n)$ is also divisible by $3^{k}$ for almost all $n$.

Theorem 1.2**.**

Let $k$ be a fixed positive integer. Then $\overline{C}_{3,1}(n)$ is almost always divisible by $3^{k}$, namely,

[TABLE]

Chen, Hirschhorn and Sellers [6] showed that $\overline{C}_{3,1}(n)$ is even for all $n\geq 1$. Hence, we have the following corollary.

Corollary 1.3**.**

Let $k$ be a fixed positive integer. Then $\overline{C}_{3,1}(n)$ is almost always divisible by $2\cdot 3^{k}$.

2. Preliminaries

In this section, we recall some definitions and basic facts on modular forms. For more details, see for example [13, 9]. We first define the matrix groups

[TABLE]

[TABLE]

and

[TABLE]

where $N$ is a positive integer. A subgroup $\Gamma$ of $\text{SL}_{2}(\mathbb{Z})$ is called a congruence subgroup if $\Gamma(N)\subseteq\Gamma$ for some $N$. The smallest $N$ such that $\Gamma(N)\subseteq\Gamma$ is called the level of $\Gamma$. For example, $\Gamma_{0}(N)$ and $\Gamma_{1}(N)$ are congruence subgroups of level $N$.

Let $\mathbb{H}:=\{z\in\mathbb{C}:\text{Im}(z)>0\}$ be the upper half of the complex plane. The group

[TABLE]

acts on $\mathbb{H}$ by $\begin{bmatrix}a&b\\ c&d\end{bmatrix}z=\displaystyle\frac{az+b}{cz+d}$. We identify $\infty$ with $\displaystyle\frac{1}{0}$ and define $\begin{bmatrix}a&b\\ c&d\end{bmatrix}\displaystyle\frac{r}{s}=\displaystyle\frac{ar+bs}{cr+ds}$, where $\displaystyle\frac{r}{s}\in\mathbb{Q}\cup\{\infty\}$. This gives an action of $\text{GL}_{2}^{+}(\mathbb{R})$ on the extended upper half-plane $\mathbb{H}^{\ast}=\mathbb{H}\cup\mathbb{Q}\cup\{\infty\}$. Suppose that $\Gamma$ is a congruence subgroup of $\text{SL}_{2}(\mathbb{Z})$. A cusp of $\Gamma$ is an equivalence class in $\mathbb{P}^{1}=\mathbb{Q}\cup\{\infty\}$ under the action of $\Gamma$.

The group $\text{GL}_{2}^{+}(\mathbb{R})$ also acts on functions $f:\mathbb{H}\rightarrow\mathbb{C}$. In particular, suppose that $\gamma=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in\text{GL}_{2}^{+}(\mathbb{R})$. If $f(z)$ is a meromorphic function on $\mathbb{H}$ and $\ell$ is an integer, then define the slash operator $|_{\ell}$ by

[TABLE]

Definition 2.1**.**

Let $\Gamma$ be a congruence subgroup of level $N$. A holomorphic function $f:\mathbb{H}\rightarrow\mathbb{C}$ is called a modular form with integer weight $\ell$ on $\Gamma$ if the following hold:

1. (1)

We have

[TABLE]

for all $z\in\mathbb{H}$ and all $\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in\Gamma$. 2. (2)

If $\gamma\in\text{SL}_{2}(\mathbb{Z})$, then $(f|_{\ell}\gamma)(z)$ has a Fourier expansion of the form

[TABLE]

where $q_{N}:=e^{2\pi iz/N}$.

For a positive integer $\ell$, the complex vector space of modular forms of weight $\ell$ with respect to a congruence subgroup $\Gamma$ is denoted by $M_{\ell}(\Gamma)$.

Definition 2.2**.**

[13, Definition 1.15] If $\chi$ is a Dirichlet character modulo $N$, then we say that a modular form $f\in M_{\ell}(\Gamma_{1}(N))$ has Nebentypus character $\chi$ if

[TABLE]

for all $z\in\mathbb{H}$ and all $\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in\Gamma_{0}(N)$. The space of such modular forms is denoted by $M_{\ell}(\Gamma_{0}(N),\chi)$.

Recall that Dedekind’s eta-function $\eta(z)$ is defined by

[TABLE]

where $q:=e^{2\pi iz}$ and $z\in\mathbb{H}$. A function $f(z)$ is called an eta-quotient if it is of the form

[TABLE]

where $N$ is a positive integer and $r_{\delta}$ is an integer.

We now recall two theorems from [13, p. 18] which will be used to prove our results.

Theorem 2.3**.**

[13, Theorem 1.64]** If $f(z)=\prod_{\delta\mid N}\eta(\delta z)^{r_{\delta}}$ is an eta-quotient such that $\ell=\frac{1}{2}\sum_{\delta\mid N}r_{\delta}\in\mathbb{Z}$,

[TABLE]

and

[TABLE]

then $f(z)$ satisfies

[TABLE]

for every $\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in\Gamma_{0}(N)$. Here the character $\chi$ is defined by $\chi(d):=\left(\frac{(-1)^{\ell}s}{d}\right)$, where $s:=\prod_{\delta\mid N}\delta^{r_{\delta}}$.

Suppose that $f$ is an eta-quotient satisfying the conditions of Theorem 2.3 and that the associated weight $\ell$ is a positive integer. If $f(z)$ is holomorphic at all of the cusps of $\Gamma_{0}(N)$, then $f(z)\in M_{\ell}(\Gamma_{0}(N),\chi)$. The following theorem gives the necessary criterion for determining orders of an eta-quotient at cusps.

Theorem 2.4**.**

[13, Theorem 1.65]** Let $c,d$ and $N$ be positive integers with $d\mid N$ and $\gcd(c,d)=1$. If $f$ is an eta-quotient satisfying the conditions of Theorem 2.3 for $N$, then the order of vanishing of $f(z)$ at the cusp $\frac{c}{d}$ is

[TABLE]

3. Proof of Theorems 1.1 and 1.2

The generating function of $\overline{C}_{3,1}(n)$ is given by

[TABLE]

We note that $\eta(24z)=q\prod_{n=1}^{\infty}(1-q^{24n})$ is a power series of $q$. Given a prime $p$, let

[TABLE]

Then using binomial theorem we have

[TABLE]

Define $B_{p,k}(z)$ by

[TABLE]

Modulo $p^{k+1}$, we have

[TABLE]

Combining (3.1) and (3.4), we obtain

[TABLE]

We now give a sketch of the idea of the proof of our main results. Firstly, we show that the eta-quotient $B_{p,k}(z)$ is a modular form on $\Gamma_{0}(576)$ with some Nebentypus character for $p=2,3$. Then we apply a deep theorem of Serre regarding the divisibility of Fourier coefficients of modular forms to $B_{2,k}(z)$ and $B_{3,k}(z)$. Finally, we use the congruence $\eqref{4.3}$ to deduce divisibility properties of $\overline{C}_{3,1}(n)$.

Proof of Theorem 1.1.

We put $p=2$ in (3.3) to obtain

[TABLE]

Now, $B_{2,k}$ is an eta-quotient with $N=576$. The cusps of $\Gamma_{0}(576)$ are represented by fractions $\frac{c}{d}$ where $d\mid 576$ and $\gcd(c,d)=1$. For example, see [12, p. 5]. By Theorem 2.4, we find that $B_{2,k}(z)$ is holomorphic at a cusp $\frac{c}{d}$ if and only if

[TABLE]

Equivalently, if and only if

[TABLE]

In the following table, we find all the possible values of $D$.

[TABLE]

Since $D\geq 0$ for all $d\mid 576$, we have that $B_{2,k}(z)$ is holomorphic at every cusp $\frac{c}{d}$. Using Theorem 2.3, we find that the weight of $B_{2,k}(z)$ is $\ell=2^{k-1}$. Also, the associated character for $B_{2,k}(z)$ is given by $\chi_{1}=(\frac{(-1)^{\ell}2^{2^{k+1}}3^{2^{k}+1}}{\bullet})$. Finally, Theorem 2.3 yields that $B_{2,k}(z)\in M_{2^{k-1}}\left(\Gamma_{0}(576),\chi_{1}\right)$. Let $m$ be a positive integer. By a deep theorem of Serre [13, p. 43], if $f(z)\in M_{\ell}(\Gamma_{0}(N),\chi)$ has Fourier expansion

[TABLE]

then there is a constant $\alpha>0$ such that

[TABLE]

Since $B_{2,k}(z)\in M_{2^{k-1}}\left(\Gamma_{0}(576),\chi_{1}\right)$, the Fourier coefficients of $B_{2,k}(z)$ are almost always divisible by $m=2^{k}$. Now, using (3.5) we complete the proof of the theorem. ∎

Proof of Theorem 1.2.

We put $p=3$ in (3.3) to obtain

[TABLE]

As before, the cusps of $\Gamma_{0}(576)$ are represented by fractions $\frac{c}{d}$ where $d\mid 576$ and $\gcd(c,d)=1$. By Theorem 2.4, $B_{3,k}(z)$ is holomorphic at a cusp $\frac{c}{d}$ if and only if

[TABLE]

Equivalently, if and only if

[TABLE]

Using the table used to evaluate the values of $D$, we find that $L\geq 0$ for all $d\mid 576$. As before, using Theorem 2.3 we find that $B_{3,k}(z)\in M_{3^{k}}\left(\Gamma_{0}(576),\chi_{2}\right)$, where the character $\chi_{2}$ is given by $\chi_{2}=(\frac{-2^{2\cdot 3^{k+1}}3^{3^{k}+1}}{\bullet})$. Using the same reasoning and (3.5), we find that $\overline{C}_{3,1}(n)$ is divisible by $3^{k}$ for almost all $n\geq 0$. This completes the proof of the theorem. ∎