1·1. Rings of motives

Let $K_0(St_{{\mathbb{C}}})$ be the Grothendieck ring of stacks. It can be defined as the localisation of the ordinary Grothendieck ring of varieties $K_0(\text{Var}_{{\mathbb{C}}})$ at the classes $[\textrm{GL}_k]$ of general linear groups [Reference Bridgeland4]. The invariants we want to study will live in the extended ring

\begin{equation*}{\mathcal M}_{{\mathbb{C}}} = K_0({\textrm{St}}_{{\mathbb{C}}})\bigl[{\mathbb{L}}^{-\frac{1}{2}}\bigr],\end{equation*}

where ${\mathbb{L}} = [{\mathbb{A}}^1] \in K_0(\text{Var}_{{\mathbb{C}}}) \to K_0({\textrm{St}}_{{\mathbb{C}}})$ is the Lefschetz motive.

1·1·1. The virtual motive of a critical locus

Let U be a smooth d-dimensional ${\mathbb{C}}$ -scheme, $f\colon U \to {\mathbb{A}}^1$ a regular function. The virtual motive of the critical locus $\text{crit}\; f = Z( \text{d} f) \subset U$ , depending on the pair (U, f), is defined in [Reference Behrend, Bryan and Szendrői3] as the motivic class

\begin{equation*}\bigl[\text{crit}\; f\bigr]_{\text{vir}} = -{\mathbb{L}}^{-\frac{d}{2}}\cdot \left[\phi_f\right] \,\in \,{\mathcal M}_{{\mathbb{C}}}^{\hat\mu},\end{equation*}

where $[\phi_f] \in K_0^{\hat\mu}(\text{Var}_{{\mathbb{C}}})$ is the (absolute) motivic vanishing cycle class defined by Denef and Loeser [Reference Denef and Loeser13]. The ‘ $\hat\mu$ ’ decoration means that we are considering $\hat\mu$ -equivariant motives, where $\hat\mu$ is the group of all roots of unity. However, the motivic invariants studied here will live in the subring ${\mathcal M}_{{\mathbb{C}}}\subset {\mathcal M}_{{\mathbb{C}}}^{\hat\mu}$ of classes carrying the trivial $\hat\mu$ -action.

Example 1·1. Set $f = 0$ . Then $\text{crit}\; f = U$ and $[U]_{\text{vir}} = {\mathbb{L}}^{-(\dim U)/2}\cdot [U]$ . For instance, $[{\textrm{GL}}_k]_{\text{vir}} = {\mathbb{L}}^{-k^2/2}\cdot [{\textrm{GL}}_k]$ .

Remark 1·2. We use lambda-ring conventions on $\mathcal M_{{\mathbb{C}}}$ from [Reference Behrend, Bryan and Szendrői3, Reference Davison and Ricolfi11]. In particular we use the definition of $[\text{crit}\; f]_{\text{vir}}$ from [Reference Behrend, Bryan and Szendrői3, section 2.8], which differs (slightly) from the one in [Reference Morrison, Mozgovoy, Nagao and Szendröi25]. The difference amounts to the substitution ${\mathbb{L}}^{1/2}\to -{\mathbb{L}}^{1/2}$ . The Euler number specialisation with our conventions is ${\mathbb{L}}^{1/2}\to -1$ .

1·2. Quivers: framings, and motivic quantum torus

A quiver Q is a finite directed graph, determined by its sets $Q_0$ and $Q_1$ of vertices and edges, respectively, along with the maps h, $t\colon Q_1 \to Q_0$ specifying where an edge starts or ends. We use the notation

to denote the tail and the head of an edge $a \in Q_1$ .

All quivers in this paper will be assumed connected. The path algebra ${\mathbb{C}} Q$ of a quiver Q is defined, as a ${\mathbb{C}}$ -vector space, by using as a ${\mathbb{C}}$ -basis the set of all paths in the quiver, including a trivial path $\epsilon_i$ for each $i \in Q_0$ . The product is defined by concatenation of paths whenever the operation is possible, and is set to be 0 otherwise. The identity element is $\sum_{i\in Q_0}\epsilon_i \in {\mathbb{C}} Q$ .

On a quiver Q one can define the Euler–Ringel form $\chi_Q(-,-)\colon {\mathbb{Z}}^{Q_0}\times {\mathbb{Z}}^{Q_0} \to {\mathbb{Z}}$ by

\begin{equation*}\chi_Q(\alpha,\beta) = \sum_{i \in Q_0}\alpha_i\beta_i - \sum_{a \in Q_1}\alpha_{t(a)}\beta_{h(a)},\end{equation*}

as well as the skew-symmetric form

\begin{equation*}{\langle\alpha,\beta\rangle}_Q = \chi_Q(\alpha,\beta)-\chi_Q(\beta,\alpha).\end{equation*}

Let Q be a quiver. Define its motivic quantum torus (or twisted motivic algebra) as

\begin{equation*}\mathcal T_Q = \prod_{\alpha \in {\mathbb{N}}^{Q_0}} \mathcal{M}_{{\mathbb{C}}}\cdot y^\alpha\end{equation*}

with product

(1·1) \begin{equation} y^\alpha\cdot y^\beta = {\mathbb{L}}^{\frac{1}{2}{\langle\alpha,\beta\rangle}_Q}y^{\alpha+\beta}.\end{equation}

If $\widetilde{Q}$ is the r-framed quiver associated to (Q, 0), one has a decomposition

\begin{equation*}\mathcal T_{\widetilde{Q}} = \left(\prod_{d\geq 0} {\mathcal M}_{{\mathbb{C}}}\cdot y_\infty^d\right) \oplus \mathcal T_Q ,\end{equation*}

where we have set $y_\infty = y^{(1,\textbf{0})}$ . A generator $y^\alpha \in \mathcal T_Q$ will be identified with its image $y^{(0,\alpha)} \in \mathcal T_{\widetilde{Q}}$ .

1·3. Quiver representations and their stability

Let Q be a quiver. A representation $\rho$ of Q is the datum of a finite dimensional ${\mathbb{C}}$ -vector space $\rho_i$ for every vertex $i\in Q_0$ , and a linear map $\rho(a)\colon \rho_i\to \rho_j$ for every edge $a\colon i\to j$ in $Q_1$ . The dimension vector of $\rho$ is $\underline{\dim}\,\rho = (\dim_{{\mathbb{C}}} \rho_i)_i\in \mathbb N^{Q_0}$ , where ${\mathbb{N}}={\mathbb{Z}}_{\geq 0}$ .

The space of all representations of Q with a fixed dimension vector $\alpha\in \mathbb N^{Q_0}$ is the affine space

\begin{equation*}\textrm{R}(Q,\alpha) = \prod_{a \in Q_1}\text{Hom}_{{\mathbb{C}}}({\mathbb{C}}^{\alpha_{t(a)}},{\mathbb{C}}^{\alpha_{h(a)}}).\end{equation*}

The gauge group ${\textrm{GL}}_\alpha = \prod_{i\in Q_0} {\textrm{GL}}_{\alpha_i}$ acts on $\textrm{R}(Q,\alpha)$ by $(g_i)_i \cdot (\rho(a))_{a\in Q_1} = (g_{h(a)}\circ\rho(a)\circ g_{t(a)}^{-1})_{a \in Q_1}$ .

Following [Reference Morrison, Mozgovoy, Nagao and Szendröi25], we recall the notion of (semi)stability of a representation.

Note that every vector $\zeta\in {\mathbb{R}}^{Q_0}$ induces a central charge $\text{Z}_{\zeta}$ if we set $\text{Z}_{\zeta}(\alpha) = -\zeta\cdot \alpha + \lvert\alpha\rvert\sqrt{-1}$ , where $\lvert\alpha\rvert = \sum_{i \in Q_0}\alpha_i$ . We denote by $\varphi_\zeta$ the induced phase function, and we set $\varphi_\zeta(\rho) = \varphi_\zeta(\underline{\dim}\,\rho)$ for every representation $\rho$ of Q.

Definition 1·5. Fix $\zeta\in {\mathbb{R}}^{Q_0}$ . A representation $\rho$ of Q is called $\zeta$ -semistable if

\begin{equation*}\varphi_\zeta(\rho')\leq \varphi_\zeta(\rho)\end{equation*}

for every nonzero proper subrepresentation $0\neq \rho'\subsetneq \rho$ . If ‘ $\leq$ ’ can be replaced by ‘ $<$ ’, we say that $\rho$ is $\zeta$ -stable. Vectors $\zeta\in {\mathbb{R}}^{Q_0}$ are referred to as stability parameters.

The sets of $\zeta$ -stable and $\zeta$ -semistable representations with given dimension vector $\alpha$ form a chain of open subsets

\begin{equation*}\textrm{R}^{\zeta\text{-st}}(Q,\alpha)\subset \textrm{R}^{\zeta\text{-ss}}(Q,\alpha)\subset \textrm{R}(Q,\alpha).\end{equation*}

If $\zeta$ is $\alpha$ -generic, one has $\textrm{R}^{\zeta\text{-st}}(Q,\alpha) = \textrm{R}^{\zeta\text{-ss}}(Q,\alpha)$ .

1·4. Quivers with potential

Let Q be a quiver, and let ${\mathbb{C}} Q$ be the path algebra of Q. A finite linear combination of cyclic paths $W \in {\mathbb{C}} Q$ is called a potential. Given a cyclic path w and an arrow $a \in Q_1$ , one defines the noncommutative derivative

\begin{equation*}\frac{\partial w}{\partial a} =\sum_{\substack{w=cac' \\ c,c'\text{ paths in }Q}}c'c\,\in\,{\mathbb{C}} Q.\end{equation*}

This rule extends to an operator $\partial/\partial a$ acting on every potential. The Jacobi algebra $J=J_{Q,W}$ of (Q, W) is the quotient of ${\mathbb{C}} Q$ by the two-sided ideal generated by $\partial W/\partial a$ for all edges $a \in Q_1$ . For every $\alpha \in {\mathbb{N}}^{Q_0}$ , the potential $W = \sum_ca_cc$ determines a regular function

\begin{equation*}f_\alpha \colon \textrm{R}(Q,\alpha) \to {\mathbb{A}}^1,\quad \rho \longmapsto \sum_{c\text{ cycle in }Q}a_c {\textrm{Tr}} (\rho(c)).\end{equation*}

The points in the critical locus $\text{crit}\; f_\alpha \subset R(Q,\alpha)$ correspond to $\alpha$ -dimensional J-modules.

Fix an $\alpha$ -generic stability parameter $\zeta \in {\mathbb{R}}^{Q_0}$ . If $f_{\zeta,\alpha} \colon \textrm{R}^{\zeta\text{-st}}(Q,\alpha)\to {\mathbb{A}}^1$ is the restriction of $f_\alpha$ , then

\begin{equation*}\mathfrak M(J,\alpha) = [\text{crit}\; f_\alpha/\textrm{G}_\alpha],\quad \mathfrak M_\zeta(J,\alpha) = [\text{crit}\; f_{\zeta,\alpha} / {\textrm{GL}}_\alpha]\end{equation*}

are, by definition, the stacks of $\alpha$ -dimensional J-modules and $\zeta$ -stable J-modules.

From now on we assume (Q, W) admits a cut. This condition ensures that the motive $[\mathfrak M(J,\alpha)]_{\text{vir}}$ introduced in the next definition lives in $\mathcal M_{{\mathbb{C}}}$ . The 3-loop quiver $L_3$ with the cubic potential $W=A_3[A_1,A_2]$ has $I=\{A_3\}$ as a cut. We go back to this in Example 1·13.

Definition 1·8 ([Reference Morrison, Mozgovoy, Nagao and Szendröi25]). We define motivic Donaldson–Thomas invariants

\begin{equation*} \bigl[\mathfrak M(J,\alpha)\bigr]_{\text{vir}} \,=\, \frac{\left[\text{crit}\; f_\alpha\right]_{\text{vir}}}{\left[{\textrm{GL}}_\alpha\right]_{\text{vir}}},\qquad \left[\mathfrak M_\zeta(J,\alpha)\right]_{\text{vir}} \,=\, {\mathbb{L}}^{\frac{1}{2}\chi(\alpha,\alpha)}\frac{\bigl[f_{\zeta,\alpha}^{-1}(0)\bigr] - \bigl[f_{\zeta,\alpha}^{-1}(1)\bigr]}{\left[{\textrm{GL}}_\alpha\right]}\end{equation*}

in ${\mathcal M}_{{\mathbb{C}}}$ , where $[{\textrm{GL}}_\alpha]_{\text{vir}}$ is taken as in Example 1·1. The generating function

(1·2) \begin{equation}A_U = \sum_{\alpha \in {\mathbb{N}}^{Q_0}}\,\bigl[\mathfrak M(J,\alpha)\bigr]_{\text{vir}}\cdot y^\alpha \,\in\,{\mathcal T}_{Q}\end{equation}

is called the universal series of (Q, W).

By [Reference Morrison, Mozgovoy, Nagao and Szendröi25, lemma 1.9], since we assumed the existence of a cut, we have an identity $ \left[\mathfrak M_\zeta(J,\alpha)\right]_{\text{vir}} = [\text{crit}\; f_{\zeta,\alpha}]_{\text{vir}}/[\textrm{G}_\alpha]_{\text{vir}}$ as long as $\alpha$ has 1 as one of its components.

1·5. Framed motivic DT invariants

Let Q be a quiver, $r\geq 1$ be an integer, and consider its r-framing $\widetilde Q$ with respect to a vertex $0 \in Q_0$ (Definition 1·3). A representation $\widetilde \rho$ of $\widetilde Q$ can be uniquely written as a pair $(u,\rho)$ , where $\rho$ is a representation of Q and $u = (u_1,\dots,u_r)$ is an r-tuple of linear maps $u_i\colon \widetilde{\rho}_\infty\to \rho_0$ . From now on, we assume our framed representations to satisfy $\dim_{{\mathbb{C}}} \widetilde{\rho}_\infty = 1$ , so that according to Convention 1·1 we can write $\underline{\dim}\,\widetilde \rho=(1,\underline{\dim}\,\rho)$ . We also view $\rho$ as a subrepresentation of $\widetilde{\rho}$ of dimension $(0,\underline{\dim}\,\rho)$ , based at the vertex $0 \in Q_0$ .

We now define motivic DT invariants for moduli stacks of r-framed representations of a given quiver Q. Fix a potential W on Q. Let $\widetilde{Q}$ be the r-framing of Q at a given vertex $0 \in Q_0$ , and let $\widetilde J$ be the Jacobi algebra $J_{\widetilde{Q},W}$ , where W is viewed as a potential on $\widetilde Q$ in the obvious way. For a generic stability parameter $\zeta \in {\mathbb{R}}^{Q_0}$ , and an arbitrary dimension vector $\alpha \in {\mathbb{N}}^{Q_0}$ , set

\begin{equation*}\zeta_\infty = -\zeta\cdot \alpha,\quad \widetilde \zeta = (\zeta_\infty,\zeta),\quad \widetilde \alpha = (1,\alpha).\end{equation*}

As in Section 1·4, consider the trace map $f_{\widetilde{\alpha}}\colon \textrm{R}(\widetilde{Q},\widetilde{\alpha})\to {\mathbb{A}}^1$ , induced by W, and its restriction to the framed-stable locus $f_{\widetilde{\zeta},\widetilde{\alpha}}\colon \textrm{R}^{\widetilde{\zeta}\text{-st}}(\widetilde{Q},\widetilde{\alpha}) \to {\mathbb{A}}^1$ . Define the moduli stacks

\begin{equation*}\mathfrak M(\widetilde J,\alpha) = \left[\text{crit}\; f_{\widetilde{\alpha}} \,\big/ {\textrm{GL}}_\alpha\right],\quad \mathfrak M_\zeta (\widetilde J,\alpha) = \left[\text{crit}\; f_{\widetilde{\zeta},\widetilde{\alpha}} \,\big/ {\textrm{GL}}_\alpha\right].\end{equation*}

Note that we are not quotienting by ${\textrm{GL}}_{\widetilde{\alpha}} = {\textrm{GL}}_\alpha \times {\mathbb{C}}^\times$ , but only by ${\textrm{GL}}_\alpha$ .

Definition 1·11. (See [Reference Morrison, Mozgovoy, Nagao and Szendröi25] for $r=1$ ) We define r-framed motivic Donaldson–Thomas invariants

(1·3) \begin{equation} \left[\mathfrak M(\widetilde J,\alpha) \right]_{\text{vir}} \,=\,\frac{\left[\text{crit}\; f_{\widetilde\alpha}\right]_{\text{vir}}}{\left[{\textrm{GL}}_{\alpha}\right]_{\text{vir}}},\qquad \left[\mathfrak M_\zeta(\widetilde J,\alpha) \right]_{\text{vir}} \,=\,\frac{\bigl[\text{crit}\; f_{\widetilde \zeta,\widetilde\alpha}\bigr]_{\text{vir}}}{\left[{\textrm{GL}}_{\alpha}\right]_{\text{vir}}}\end{equation}

in $\mathcal M_{{\mathbb{C}}}$ , and the associated motivic generating functions

\begin{align*}\widetilde A_U&= \sum_{\alpha \in {\mathbb{N}}^{Q_0}} \,\left[\mathfrak M(\widetilde J,\alpha) \right]_{\text{vir}}\cdot y^{\widetilde\alpha} \in \mathcal T_{\widetilde Q} \\\mathsf Z_\zeta&= \sum_{\alpha \in {\mathbb{N}}^{Q_0}} \,\left[\mathfrak M_\zeta(\widetilde J,\alpha) \right]_{\text{vir}}\cdot y^{\widetilde\alpha} \in \mathcal T_{\widetilde Q}.\end{align*}

The fact that the r-framed invariants (1·3) live in $\mathcal M_{{\mathbb{C}}}$ (i.e. have no monodromy) follows from [Reference Morrison, Mozgovoy, Nagao and Szendröi25, Lemma 1·10]. The reason is that the dimension vector $\widetilde{\alpha} = (1,\alpha)$ contains ‘1’ as a component.

1·6. Dimensional reduction

If I is a cut for (Q, W), one can define a new quiver $Q_I = (Q_0,Q_1\setminus I)$ . Let $J_{W,I}$ be the quotient of ${\mathbb{C}} Q_I$ by the two-sided ideal generated by the noncommutative derivatives $\partial W/\partial a$ for $a \in I$ . Let $\textrm{R}(J_{W,I},\alpha) \subset \textrm{R}(Q_I,\alpha)$ be the space of $J_{W,I}$ -modules of dimension vector $\alpha$ . Then one has the following dimensional reduction principle.

Proposition 1·12. ([Reference Morrison, Mozgovoy, Nagao and Szendröi25, proposition 1·15]) Suppose I is a cut for (Q, W). Set $\textrm{d}_I(\alpha) = \sum_{a \in I}\alpha_{t(a)}\alpha_{h(a)}$ . Then

\begin{equation*}A_U = \sum_{\alpha \in {\mathbb{N}}^{Q_0}}{\mathbb{L}}^{\frac{1}{2}\chi_Q(\alpha,\alpha) + \textrm{d}_I(\alpha)}\frac{[{\textrm{R}}(J_{W,I},\alpha)]}{[{\textrm{GL}}_\alpha]} \cdot y^\alpha.\end{equation*}

Example 1·13. Let $Q=L_3$ be the 3-loop quiver (see Figure 1, and remove the framing vertex to obtain a picture of this quiver) with the potential $W=A_3[A_1,A_2]$ . Notice that $J=J_{L_3,W}={\mathbb{C}}[x,y,z]$ , and $I = \{A_3\}$ is a cut for $(L_3,W)$ . The quiver $Q_I$ is the 2-loop quiver and $J_{W,I} = {\mathbb{C}}[x,y]$ . We have $\text{d}_I(n)=n^2$ and $\chi_Q(n,n)=-2n^2$ . Therefore Proposition 1·12 yields an identity

(1·4) \begin{equation}\sum_{n\geq 0}\,\bigl[\mathfrak M(J,n) \bigr]_{\text{vir}}\cdot y^n = \sum_{n\geq 0}\frac{[C_n]}{[{\textrm{GL}}_n]}\cdot y^n = \prod_{m \geq 1}\prod_{k \geq 1}\left(1-{\mathbb{L}}^{2-k}y^m\right)^{-1},\end{equation}

where $R(J_{W,I},n)$ is identified with the commuting variety

\begin{equation*}C_n = \{(A_1,A_2) \in \text{End}_{{\mathbb{C}}}({\mathbb{C}}^n)^{\oplus 2} | [A_1,A_2] = 0\} \subset \text{End}_{{\mathbb{C}}}({\mathbb{C}}^n)^{\oplus 2},\end{equation*}

and the second identity in (1·4) is the Feit–Fine formula [Reference Behrend, Bryan and Szendrői3, Reference Bryan and Morrison5, Reference Feit and Fine15].

Fig. 1.

The r-framed 3-loop quiver $\widetilde{L}_3$ .

Fig. 2.

An illustration of Proposition 2·6.

2·1. Stability on the framed 3-loop quiver

The main character in this section is the framed quiver $\widetilde{L}_3$ of Figure 1, which we equip with the potential $W=A_3[A_1,A_2]$ .

Definition 2·1. Let $\widetilde{\rho}=(u,\rho)$ be a representation of $\widetilde{L}_3$ of dimension (1, n). We denote by $\langle{u,\rho}\rangle \subset \widetilde{\rho}$ the smallest subrepresentation of $\widetilde{\rho}$ containing $u(\rho_{\infty})$ . More precisely, if $\rho = (A_1,A_2,A_3) \in R(L_3,n)$ , then $\langle{u,\rho}\rangle$ is the subrepresentation of $\widetilde\rho$ with ${\langle{u,\rho}\rangle}_\infty = \widetilde{\rho}_\infty = {\mathbb{C}}$ and

\begin{equation*}{\langle{u,\rho}\rangle}_0 = Span_{{\mathbb{C}}}\{A_1^{a_1}A_2^{a_2}A_3^{a_3}\cdot u_\ell(1)|a_i\geq 0,1\leq \ell\leq r\} \subset \widetilde{\rho}_0,\end{equation*}

and with linear maps induced naturally by those defined by $\widetilde{\rho}$ .

From now on we identify the space of stability parameters for $L_3$ with ${\mathbb{R}}$ .

Proof. For the case $\zeta<0$ we refer to [Reference Beentjes and Ricolfi1, proposition 2·4]. Consider the case $\zeta>0$ . If we had $n = \dim_{{\mathbb{C}}}\widetilde{\rho}_0 > 0$ , then $\rho \subset \widetilde{\rho}$ would be destabilising, for $\text{Z}_{\widetilde{\zeta}}(0,n) = -n\zeta + n\sqrt{-1}$ implies $\varphi_{\widetilde{\zeta}}(\rho) > 1/2 = \varphi_{\widetilde{\zeta}}(\widetilde{\rho})$ , since $\text{Z}_{\widetilde{\zeta}}(1,n)=(n+1)\sqrt{-1}$ has vanishing real part. On the other hand, if $n = 0$ then $\widetilde{\rho}$ is simple and hence $\zeta$ -stable. In the case $\zeta=0$ there is nothing to prove, as all representations have phase $1/2$ .

Consider the following regions of the space of stability parameters ${\mathbb{R}}$ :

1. o $\Omega_{+} = \{\zeta\in {\mathbb{R}} | \zeta < 0 \}$ ,

2. o $\Omega_{0} = \{\zeta\in {\mathbb{R}} | \zeta=0 \}$ ,

3. o $\Omega_{-} = \{\zeta\in {\mathbb{R}} | \zeta > 0\}$ .

By Lemma 2·2 the space of stability parameters on $\widetilde{{L}}_3$ admits a particularly simple wall-and-chamber decomposition ${\mathbb{R}} = \Omega_{+} \amalg \Omega_{0} \amalg \Omega_{-}$ with one wall (the origin) and two chambers.

2·2. The virtual motive of the Quot scheme of points

Let $\widetilde{L}_3$ be the r-framed 3-loop quiver (Figure 1), and fix the potential $W=A_3[A_1,A_2]$ . Fix a stability parameter $\zeta^+ \in \Omega_+ = {\mathbb{R}}_{<0}$ . Fixing $n\geq 0$ and setting $\widetilde{\zeta}^+ = (-n\zeta,\zeta)$ , $\widetilde n = (1,n)$ , the quotient stack

\begin{equation*}\mathfrak M_{\zeta^+}(\widetilde{L}_3,n) = \bigl[\textrm{R}^{\widetilde{\zeta}^+\text{-st}}(\widetilde{L}_3,\widetilde n)\, \big/ {\textrm{GL}}_n\bigr]\end{equation*}

is a smooth quasiprojective variety of dimension $2n^2+rn$ , called the noncommutative Quot scheme in [Reference Beentjes and Ricolfi1]. The regular function $f_n\colon \textrm{R}(\widetilde{L}_3,\widetilde n) \to {\mathbb{A}}^1$ given by taking the trace of W descends to a regular function on $\mathfrak M_{\zeta^+}(\widetilde{L}_3,n)$ , still denoted $f_n$ . We have the following description of the Quot scheme of length n quotients of $\mathscr O_{{\mathbb{A}}^3}^{\oplus r}$ .

Proposition 2·3. ([Reference Beentjes and Ricolfi1, theorem 2·6]) There is an identity of closed subschemes

\begin{equation*}{\textrm{Quot}}_{{\mathbb{A}}^3}(\mathscr O^{\oplus r},n) = \textrm{crit}(f_n) \subset \mathfrak M_{\zeta^+}(\widetilde{L}_3,n).\end{equation*}

Thanks to Proposition 2·3, we can form the virtual motives of the Quot scheme, as in Section 1·1·1, and define their generating function

(2·1) \begin{equation}\mathsf{DT}_r^{\textrm{points}}({\mathbb{A}}^3,q) = \sum_{n\geq 0}\,\left[{\textrm{Quot}}_{{\mathbb{A}}^3}(\mathscr O^{\oplus r},n) \right]_{\text{vir}}\cdot q^n\,\in\,{\mathcal M}_{{\mathbb{C}}}[\![q]\!].\end{equation}

Remark 2·4. The main result of [Reference Behrend, Bryan and Szendrői3] is the formula

\begin{equation*}\mathsf{DT}_1^{\textrm{points}}({\mathbb{A}}^3,q) = \prod_{m\geq 1}\prod_{k=0}^{m-1}\left(1-{\mathbb{L}}^{2+k-\frac{m}{2}}q^m\right)^{-1}.\end{equation*}

The series $\mathsf{DT}_1^{\textrm{points}}(Y,q)$ studied in [Reference Behrend, Bryan and Szendrői3] for an arbitrary smooth 3-fold Y also appeared in [Reference Davison and Ricolfi11] as the wall-crossing factor in the motivic DT/PT correspondence based at a fixed smooth curve $C \subset Y$ in Y. This correspondence refined its enumerative counterpart [Reference Ricolfi32, Reference Ricolfi33]. The same phenomenon occurred in [Reference Morrison, Mozgovoy, Nagao and Szendröi25, Reference Morrison and Nagao26] in the context of framed motivic DT invariants. See [Reference Bridgeland4, Reference Ricolfi35] for a generalisation $\mathsf{DT}_r^{\textrm{points}}(Y,q)$ of (2·1) to an arbitrary smooth 3-fold Y. See [Reference Ricolfi34] for a plethystic formula expressing the naive motives $[{\textrm{Quot}}_Y(F,n)] \in K_0(\text{Var}_{{\mathbb{C}}})$ in terms of the motives of the punctual Quot schemes.

The main identity of Theorem 0·1

(2·2) \begin{equation}\mathsf{DT}_r^{\textrm{points}}({\mathbb{A}}^3,q) = \prod_{m\geq 1}\prod_{k=0}^{rm-1}\left(1-{\mathbb{L}}^{2+k-\frac{rm}{2}}q^m\right)^{-1}\end{equation}

will be proved in the next subsection. An argument following closely the technique used in the $r=1$ case by Behrend–Bryan–Szendrői [Reference Behrend, Bryan and Szendrői3], was given in the PhD theses of the first and third authors [Reference Cazzaniga6, Reference Ricolfi31]. In the following subsection, we will instead exploit an r-framed version of motivic wall-crossing. This technique, inspired by [Reference Morrison, Mozgovoy, Nagao and Szendröi25, Reference Morrison and Nagao26, Reference Mozgovoy28], will be also applied to small crepant resolutions of affine toric Calabi–Yau 3-folds in [Reference Cazzaniga and Ricolfi8].

Granting (2·2) for the moment, we prove in the next lemma the second assertion in Theorem 0·1, i.e. Formula (0·2).

Lemma 2·5. There is an identity

(2·3) \begin{equation}\prod_{m\geq 1}\prod_{k=0}^{rm-1}\left(1-{\mathbb{L}}^{2+k-\frac{rm}{2}}q^m\right)^{-1} \,=\, \prod_{i=1}^r \mathsf{DT}_1^{\textrm{points}}\left({\mathbb{A}}^3,q{\mathbb{L}}^{\frac{-r-1}{2}+i}\right).\end{equation}

Proof. The sought after identity follows from a simple manipulation:

\begin{align*} \prod_{i=1}^r\mathsf{DT}_1^{\textrm{points}}({\mathbb{A}}^3,q{\mathbb{L}}^{\frac{-r-1}{2}+i}) &\,=\,\prod_{i=1}^r \prod_{m\geq 1}\prod_{k=0}^{m-1}\left(1-{\mathbb{L}}^{2+k-\frac{m}{2}}{\mathbb{L}}^{\frac{-r-1}{2}m+im}q^m\right)^{-1} \\ &\,=\,\prod_{m\geq 1}\prod_{k=0}^{m-1}\prod_{i=1}^r \left(1-{\mathbb{L}}^{2+k+(i-1)m-\frac{rm}{2}}q^m\right)^{-1} \\ &\,=\,\prod_{m\geq 1}\prod_{k=0}^{rm-1}\left(1-{\mathbb{L}}^{2+k-\frac{rm}{2}}q^m\right)^{-1},\end{align*}

as claimed.

2·3. Calculation via wall-crossing

In this subsection we prove Theorem 0·1. Consider the universal generating function

\begin{equation*}\widetilde A_U = \sum_{n\geq 0}\,\left[\mathfrak M(\widetilde J,n) \right]_{\text{vir}}\cdot y^{(1,n)}\,\in\,\mathcal T_{\widetilde L_3}\end{equation*}

as an element of the motivic quantum torus. To (generic) stability parameters $\zeta^{\pm} \in \Omega_{\pm}$ we associate elements (cf. Definition 1·11)

\begin{equation*}\mathsf Z_{\zeta^\pm} = \sum_{n\geq 0}\,\left[\mathfrak M_{\zeta^{\pm}}(\widetilde J,n) \right]_{\text{vir}}\cdot y^{(1,n)}\,\in\,\mathcal T_{\widetilde L_3}.\end{equation*}

By Lemma 2·2 (iii), we have an identity

(2·4) \begin{equation}\mathsf Z_{\zeta^-} = y_{\infty} = y^{(1,0)},\end{equation}

whereas the series $\mathsf Z_{\zeta^+}$ is, essentially, a “shift” of the generating function $\mathsf{DT}_r^{\textrm{points}}({\mathbb{A}}^3,y^{(0,1)})$ . More precisely, by Proposition 2·3 we have an identification

\begin{equation*}\mathfrak M_{\zeta^{+}}(\widetilde J,n) = {\textrm{Quot}}_{{\mathbb{A}}^3}(\mathscr O^{\oplus r},n) \subset \mathfrak M_{\zeta^+}(\widetilde{L}_3,n)\end{equation*}

of critical loci sitting inside the noncommutative Quot scheme; in particular, the associated virtual motives are the same. The shift is intended in the following sense: since ${\langle{(0,n),(1,0)}\rangle}_{\widetilde{L}_3} = rn$ , the product rule (1·1) yields the identity

(2·5) \begin{equation} y^{(1,n)} = {\mathbb{L}}^{-\frac{rn}{2}} y^{(0,n)}\cdot y_{\infty}\,\in\,\mathcal T_{\widetilde{L}_3}.\end{equation}

Since we can express $y^{(0,n)}$ as the n-fold product of $y^{(0,1)}$ with itself, we obtain

(2·6) \begin{equation}\mathsf Z_{\zeta^+} = \mathsf{DT}_r^{\textrm{points}}\left({\mathbb{A}}^3,{\mathbb{L}}^{-\frac{r}{2}} y^{(0,1)}\right)\cdot y_{\infty}\cdot\end{equation}

The last generating function we need to analyse is

\begin{equation*}A_U = \sum_{n\geq 0}\,\bigl[\mathfrak{M}(J,n) \bigr]_{\text{vir}} \cdot y^{(0,n)}\,\in\,\mathcal T_{L_3}\,\subset\, \mathcal T_{\widetilde{L}_3},\end{equation*}

whose n-th coefficient is the virtual motive of the stack of 0-dimensional ${\mathbb{C}}[x,y,z]$ -modules of length n. This was already computed in Example 1·13:

(2·7) \begin{equation}A_U(y) = \prod_{m \geq 1}\prod_{k \geq 1}\left(1-{\mathbb{L}}^{2-k}y^m\right)^{-1}.\end{equation}

Proposition 2·6. In $\mathcal T_{\widetilde{L}_3}$ , there are identities

(2·8) \begin{equation}\mathsf Z_{\zeta^+}\cdot A_U = \widetilde A_U = A_U \cdot \mathsf Z_{\zeta^-}.\end{equation}

Proof. Let $\widetilde{\rho} = (u,\rho)$ be a $\widetilde{J}$ -module. Consider $\zeta^+\in \Omega_{+}$ , and let $\langle{u,\rho}\rangle\subset \widetilde{\rho}$ be the submodule introduced in Definition 2·1. We have that $\langle{u,\rho}\rangle$ is $\zeta^+$ -stable by Lemma 2·2(i) and the quotient $\widetilde{\rho}/\langle{u,\rho}\rangle$ is supported at the vertex 0. From this we deduce the decomposition $\widetilde{A}_U=\mathsf Z_{\zeta^+}\cdot A_{U}$ .

Consider now $\zeta^- \in \Omega_{-}$ . The quotient of $\widetilde{\rho}$ by the submodule $\rho$ based at the vertex 0 is the simple module supported at the framing vertex $\infty$ . By Lemma 2·2 (iii) this is the unique $\zeta^-$ -stable module for the current choice of $\zeta^-$ , so we obtain the decomposition $\widetilde{A}_U=A_{U}\cdot \mathsf Z_{\zeta^-}$ .

Note that we have an identity

\begin{equation*}y_{\infty}\cdot y^{(0,n)} = {\mathbb{L}}^{-\frac{rn}{2}}\cdot y^{(1,n)} = {\mathbb{L}}^{-rn}\cdot y^{(0,n)}\cdot y_{\infty},\end{equation*}

where we have used (2·5) for the second equality. By Formula (2·6), the left-hand term of Formula (2·8) can then be rewritten as

\begin{equation*}\mathsf{DT}_r^{\textrm{points}}\left({\mathbb{A}}^3,{\mathbb{L}}^{-\frac{r}{2}} y^{(0,1)}\right)\cdot \sum_{n\geq 0}\,\bigl[\mathfrak{M}(J,n)\bigr]_{\text{vir}}\cdot y_{\infty} \cdot y^{(0,n)} \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\mathsf{DT}_r^{\textrm{points}}\left({\mathbb{A}}^3,{\mathbb{L}}^{-\frac{r}{2}} y^{(0,1)}\right)\cdot \sum_{n\geq 0}\,\bigl[\mathfrak{M}(J,n)\bigr]_{\text{vir}} {\mathbb{L}}^{-rn}\cdot y^{(0,n)}\cdot y_{\infty}.\end{equation*}

Therefore, by Equation (2·4), identifying the two expressions for $\widetilde{A}_U$ in Equation (2·8) yields

\begin{equation*}\mathsf{DT}_r^{\textrm{points}}\left({\mathbb{A}}^3,{\mathbb{L}}^{-\frac{r}{2}} y^{(0,1)}\right)\cdot A_U\left({\mathbb{L}}^{-r}y^{(0,1)}\right) \cdot y_{\infty} = A_U\left(y^{(0,1)}\right)\cdot y_{\infty},\end{equation*}

which is equivalent to

\begin{equation*}\mathsf{DT}_r^{\textrm{points}}\left({\mathbb{A}}^3,{\mathbb{L}}^{-\frac{r}{2}} y^{(0,1)}\right) = \frac{A_U\left(y^{(0,1)}\right)}{A_U\left({\mathbb{L}}^{-r}y^{(0,1)}\right)}.\end{equation*}

Setting $q = {\mathbb{L}}^{-\frac{r}{2}}y^{(0,1)}$ , and using Equation (2·7), a simple substitution yields

\begin{align*} \mathsf{DT}_r^{\textrm{points}}({\mathbb{A}}^3,q) &= \frac{A_U\left({\mathbb{L}}^{\frac{r}{2}}q\right)}{A_U\left({\mathbb{L}}^{-\frac{r}{2}}q\right)} \\ &=\prod_{m\geq 1}\prod_{j\geq 0}\frac{\left(1-\mathbb L^{1-j+\frac{rm}{2}}q^m\right)^{-1}}{\left(1-\mathbb L^{1-j-\frac{rm}{2}}q^m\right)^{-1}}\\&=\prod_{m\geq 1}\prod_{j=0}^{rm-1}\left(1-\mathbb L^{1-j+\frac{rm}{2}}q^m\right)^{-1}\\&=\prod_{m\geq 1}\prod_{k=0}^{rm-1}\left(1-\mathbb L^{2+k-\frac{rm}{2}}q^m\right)^{-1}.\end{align*}

Formula (0·1) is proved. By Lemma 2·5, Theorem 0·1 is proved.

Remark 2·8. The generating function $\mathsf{DT}_r^{\textrm{points}}({\mathbb{A}}^3,(-1)^rq)$ admits the plethystic expression

\begin{equation*}\mathsf{DT}_r^{\textrm{points}}({\mathbb{A}}^3,(-1)^rq) =Exp\left(\frac{(-1)^rq\mathbb L^{\frac{3}{2}}}{\bigl(1-(-\mathbb L^{-\frac{1}{2}})^rq\bigr)\bigl(1-(-\mathbb L^{\frac{1}{2}})^rq\bigr)}\frac{\mathbb L^{-\frac{r}{2}}-\mathbb L^{\frac{r}{2}}}{\mathbb L^{-\frac{1}{2}}-\mathbb L^{\frac{1}{2}}}\right).\end{equation*}

This is exploited in [Reference Ricolfi35, section Reference Bridgeland4] to define $[{\textrm{Quot}}_Y(F,n)]_{\text{vir}}$ for all 3-folds Y and locally free sheaves F over Y. The shift by the sign $(\!-1)^r$ is needed in order to write the series as a plethystic exponential (and cannot be naively removed from both sides). This is the case for K-theoretic DT invariants as well [Reference Fasola, Monavari and Ricolfi14, theorem A].

Remark 2·9. The Euler number specialisation ${\mathbb{L}}^{1/2} \to -1$ applied to Formula (0·1) yields a formula for the generating function of virtual Euler characteristics of Quot schemes,

\begin{equation*}\sum_{n \geq 0} \chi_{\text{vir}}({\textrm{Quot}}_{{\mathbb{A}}^3}(\mathscr O^{\oplus r},n))\cdot q^n = \mathsf M((-1)^rq)^r,\end{equation*}

where $\mathsf M(q) = \prod_{m\geq 1}(1-q^m)^{-m}$ is the MacMahon function, the generating function of plane partitions, and where $\chi_{\text{vir}}(-) = \chi(-,\nu)$ is the Euler characteristic weighted by Behrend’s microlocal function [Reference Behrend2]. The above identity can be seen as the Calabi–Yau specialisation (i.e. the specialisation $s_1+s_2+s_3 = 0$ ) of the generating function of cohomological Donaldson–Thomas invariants of ${\mathbb{A}}^3$ ,

\begin{equation*}\sum_{n\geq 0}\mathsf{DT}_r^{\textrm{coh}}({\mathbb{A}}^3,n)\cdot q^n = \mathsf M((-1)^rq)^{-r\frac{(s_1+s_2)(s_1+s_3)(s_2+s_3)}{s_1s_2s_3}}\,\,\in\,\, {\mathbb{Q}}(\!(s_1,s_2,s_3)\!) \unicode{x27E6}{q}\unicode{x27E7},\end{equation*}

obtained in [Reference Fasola, Monavari and Ricolfi14, theorem B] as a higher rank version of [Reference Maulik, Nekrasov, Okounkov and Pandharipande22, theorem 1], where $s_1$ , $s_2$ and $s_3$ are the equivariant parameters of the torus ${\mathbb{T}} = {\mathbb{G}}_m^3$ acting on the Quot scheme.

3·1. Random variables on r-coloured plane partitions

We introduce a multivariate function

(3·1) \begin{equation} F(u,v,w,z)=\prod_{l=1}^{r}\prod_{m\geq 1}\prod_{k=1}^{m}\left(1-wu^{k}v^{ml}z^m\right)^{-1}.\end{equation}

The coefficient of $z^n$ is a polynomial on the three variables u, v and w, which we denote by $Q_{n}(u,v,w)$ , whose coefficients are nonnegative integers. When $u=v=w=1$ , we obtain the well known MacMahon function raised to the power r, which is the generating function for the r-tuples of plane partitions. Hence, $Q_{n}(1,1,1)$ is the number of r-tuples of plane partitions of total size n, i.e. the number of r-coloured plane partitions $(\pi_1,\, \pi_2, \, \dots, \, \pi_r)$ such that $\sum_{j=1}^{r}\lvert\pi_j\rvert=n$ , where $\lvert\pi_j\rvert$ denotes the sum of the entries of the plane partition $\pi_j$ (or the number of boxes, cf. Figure 3). The polynomial $Q_{n}(u,v,w)$ , when divided by $Q_{n}(1,1,1)$ , represents the joint probability generating function of some random variables $X_n$ , $Y_n$ , and $Z_n$ on the space of r-coloured plane partitions of size n, where each r-tuple is equally likely. More precisely, we have

(3·2) \begin{equation}\frac{Q_{n}(u,v,w)}{Q_{n}(1,1,1)}=\mathbb{E}\left(u^{X_n}v^{Y_n}w^{Z_n}\right).\end{equation}

To describe these random variables, we need to define certain parameters of plane partitions. For a plane partition $\pi$ , let $\Delta(\pi)$ denote the sum of the diagonal parts of $\pi$ , $\Delta_{+}(\pi)$ denote the sum of the upper diagonal parts, and $\Delta_{-}(\pi)$ denote the sum of the lower diagonal parts. See Figure 3 for an example of a plane partition showing the values of these parameters.

Fig. 3.

A plane partition $\pi$ of size $\lvert\pi\rvert=31$ , $\Delta(\pi)=9$ , $\Delta_{+}(\pi)=12$ , and $\Delta_{-}(\pi)=10$ .

The parameter $\Delta(\pi)$ is also known as the trace of $\pi$ and it has been studied in the literature, see for instance [Reference Kamenov and Mutafchiev19] and the references therein. In particular, one has

\begin{equation*}\prod_{m\geq 1}(1-wz^m)^{-m}=\sum_{\pi}w^{\Delta(\pi)}z^{\lvert\pi\rvert}.\end{equation*}

Similarly, we can find in [Reference Morrison24] that

\begin{equation*}\prod_{m\geq 1}\prod_{k=1}^{m}(1-q^{2k-m}z^{m})^{-1}=\sum_{\pi}q^{\Delta(\pi)+\Delta_{+}(\pi)-\Delta_{-}(\pi)}z^{\lvert\pi\rvert}.\end{equation*}

We can easily deduce from these two identities that for an r-coloured plane partition $\overline{\pi} = (\pi_1, \pi_2, \ldots, \pi_r)$ of total size n, we have

\begin{align*} X_n\left(\overline{\pi}\right)& =\frac{n}{2}+\frac{1}{2}\left( \sum_{l=1}^{r}\left(\Delta(\pi_l)+\Delta_{+}(\pi_l)-\Delta_{-}(\pi_l)\right)\right)\\ &=\sum_{l=1}^{r}\left(\Delta(\pi_l)+\Delta_{+}(\pi_l)\right),\\ Y_n\left(\overline{\pi}\right)& = \sum_{l=1}^{r}l|\pi_l|, \ \ \text{and}\\ Z_n\left(\overline{\pi}\right)& = \sum_{l=1}^{r}\Delta(\pi_l).\end{align*}

When $r=1$ , Kamenov and Mutafchiev [Reference Kamenov and Mutafchiev19] proved that the distribution of the trace of a random plane partition of size n, when suitably normalised, is asymptotically normal. Morrison [Reference Morrison24] also established asymptotic normality for any random variable of the form $\delta\Delta(\pi)+\Delta_{+}(\pi)-\Delta_{-}(\pi)$ , where $\pi$ is a random plane partition of size n and $\delta$ is a fixed real number. We show that for any fixed integer $r\geq 1$ , any nontrivial linear combination of the variables $X_n$ , $Y_n$ and $Z_n$ , when suitably normalised, converges weakly to a normal distribution. It is worth noting that the random variable $Y_n$ is non-constant only when $r>1$ .

Proposition 3·1. For any fixed real vector $(\alpha,\beta,\gamma)\neq (0,0,0)$ , there exist sequences of real numbers $\mu_n$ and $\sigma_n\geq 0$ such that the normalised random variable

\begin{equation*}\frac{\alpha X_n+\beta Y_n+ \gamma Z_n-\mu_n}{\sigma_n}\end{equation*}

converges weakly to the standard normal distribution. Moreover, $\mu_n$ and $\sigma_n$ satisfy the following asymptotic formulas as $n\to \infty$ :

\begin{align*}\mu_n&\,\,=\,\,\left(\frac{1}{2}\alpha+\frac{r+1}{2} \beta\right)n+\frac{r^{1/3}\zeta(2)(\alpha+2\gamma)}{2^{5/3}(\zeta(3))^{2/3}}\, n^{2/3}+\mathcal{O}(n^{1/3}), \, \text{ and }\,\\\sigma_n^2 &\,\,\sim\,\,\begin{cases}\displaystyle \frac{\alpha^2+(r^2-1)\beta^2}{2^{7/3}(r\zeta(3))^{1/3}}\, n^{4/3}\, & \text{ if } (\alpha,\beta)\neq (0,0),\\[2em]\displaystyle \frac{r^{1/3}\gamma^2}{3(2\zeta(3))^{2/3}} n^{2/3}\log n \, & \text{otherwise.}\end{cases}\end{align*}

Looking at the asymptotic behaviour of the random variables $X_n$ , $Y_n$ and $Z_n$ , when divided by $n^{2/3}$ , we observe from the above result that the random variable $n^{-2/3}Z_n$ degenerates as $n\to \infty$ . Furthermore, by the Camér–Wold device [Reference Cramér and Wold9], the random variables $n^{-2/3}X_n$ , $n^{-2/3}Y_n$ converge jointly to a bivariate normal distribution with a diagonal covariance matrix. The appropriate normalisation of $Z_n$ is $n^{-1/3}(\log n)^{-1/2}Z_n$ which is, when centred, asymptotically normal. This asymptotic normality and the asymptotic formulas for $\mu_n$ and $\sigma_n^2$ agree with the main result in [Reference Kamenov and Mutafchiev19] when $r=1$ and $(\alpha,\beta,\gamma)=(0,0,1).$

Theorem 0·2 now follows immediately from Proposition 3·1 as we will see next.

3·2. Deducing Theorem 0·2

Granting Proposition 3·1, we can now finish the proof of Theorem 0·2, our second main result.

Theorem 3·2. As $n\to \infty$ , the normalised random variable $n^{-2/3}S_{n,r}$ converges in distribution to $\mathcal{N}(\mu, \sigma^2)$ with

\begin{equation*}\mu = \frac{r^{1/3}\pi^2}{2^{5/3}(\zeta(3))^{2/3}}\, \text{ and }\, \sigma^2=\frac{r^{5/3}}{(2\zeta(3))^{1/3}},\end{equation*}

where $\zeta(s)$ is Riemann’s zeta function.

Proof. With the change of variable $T={\mathbb{L}}^{1/2}$ , the combination of the equations (0·1) and (0·2) in Theorem 0·1 yields

\begin{equation*}\mathsf{DT}_r^{\textrm{points}}({\mathbb{A}}^3,q) =\prod_{l=1}^{r}\prod_{m\geq 1}\prod_{k=0}^{m-1}\left(1-T^{4+2k-m}\Big(qT^{-r-1+2l}\Big)^m\right)^{-1}.\end{equation*}

The product on the right-hand side can be expressed in terms of our auxiliary function F(u, v, w, z) defined at the beginning of this section. First we write it as follows

\begin{equation*} \prod_{l=1}^{r}\prod_{m\geq 1}\prod_{k=0}^{m-1}\left(1-T^{4+m-2(m-k)}\Big(qT^{r+1-2(r-l+1)}\Big)^m\right)^{-1}.\end{equation*}

If k goes from 0 to $m-1$ , then $m-k$ goes from m to 1. Similarly, if l goes from 1 to r, then $r-l+1$ goes from r to 1. Therefore, we have

\begin{align*}\mathsf{DT}_r^{\textrm{points}}({\mathbb{A}}^3,q)& =\prod_{l=1}^{r}\prod_{m\geq 1}\prod_{k=1}^{m}\left(1-T^{4+m-2k}(T^{r+1-2l}q)^m\right)^{-1}\\& = \prod_{l=1}^{r}\prod_{m\geq 1}\prod_{k=1}^{m}\left(1-T^{4-2k-2ml}(T^{r+2}q)^m\right)^{-1}\\& =F(T^{-2},T^{-2},T^4,T^{r+2}q).\end{align*}

This implies that

(3·3) \begin{equation}{\mathbb{E}}(T^{S_{n,r}})= \frac{M_{n,r}(T)}{M_{n,r}(1)}={\mathbb{E}}\left(T^{4Z_n-2X_n-2Y_n+(r+2)n}\right),\end{equation}

where $M_{n,r}(T)$ is, as defined in (0·3), the coefficient of $q^n$ in $\mathsf{DT}_r^{\textrm{points}}({\mathbb{A}}^3,q)$ . The second equality in (3·3) makes use of Equation (3·2). Thus, $S_{n,r}$ has the same distribution as $4Z_n-2X_n-2Y_n+(r+2)n$ — a shifted linear combination of the variables $X_n$ , $Y_n$ , and $Z_n$ . Now, applying Proposition 3·1 with $(\alpha, \beta, \gamma) = (-2, -2, 4)$ , we deduce that the normalised random variable $n^{-2/3}(4Z_n-2X_n-2Y_n+(r+2)n)$ converges weakly to the normal distribution $\mathcal{N}(\mu,\sigma^2)$ with

\begin{equation*}\mu = \frac{r^{1/3}\pi^2}{2^{5/3}(\zeta(3))^{2/3}}\, \text{ and }\, \sigma^2=\frac{r^{5/3}}{(2\zeta(3))^{1/3}},\end{equation*}

which proves Theorem 0·2.

3·3. Proof of Proposition 3·1

Morrison used the method of moments to prove his result in [Reference Morrison24]. However, due to the appearance of the second variable $Y_n$ and the complication that comes with it, we decided to use a different approach. We follow the method that Hwang used in [Reference Hwang18] to prove limit theorems for the number of parts in the so-called restricted partitions (these are one dimensional partitions with some restrictions on the parts). The first part of the proof is based on the saddle-point method to get an asymptotic formula for $Q_{n}(u,v,w)$ as $n\to\infty$ , and the second is a perturbation technique to deduce the central limit theorem.

3·3·1. Saddle-point method

The goal here is to obtain an asymptotic formula for $Q_{n}(u,v,w)$ as $n\to\infty$ , where (u, v, w) is allowed to vary in a fixed real neighbourhood of (1, 1, 1). To simplify our notation, define $\Phi(t)=-\log\!(1-e^{-t})$ , and for real numbers a, b and c, we let

\begin{equation*}f(x,y)=\sum_{l=1}^{r}\sum_{m=1}^{\infty}\sum_{k=1}^{m}\Phi(xm+y(c+ak+mbl)).\end{equation*}

The function f depends on (a, b, c) but we drop this dependence for now to ease notation. Also, for a positive number $\rho$ , we make the substitution $z=e^{-\tau}$ , where $\tau=\rho+it$ . Hence,

\begin{equation*}f(\tau,\rho)=-\sum_{l=1}^{r}\sum_{m=1}^{\infty}\sum_{k=1}^{m}\log\left(1-e^{-\rho(c+ak+mbl)-\tau m}\right)=\log F(e^{-a\rho},e^{-b\rho},e^{-c\rho},e^{-\tau}).\end{equation*}

One can easily verify that if (u, v, w) is bounded (which is the case throughout this section), then there exists a fixed positive real number R such that the product in (3·1) converges absolutely whenever $|z|<R$ . Hence, F(u, v, w, z), as function of z, is analytic in a complex neighbourhood of 0. By Cauchy’s integral formula, we have

(3·4) \begin{equation}Q_{n}(e^{-a\rho},e^{-b\rho},e^{-c\rho})=\frac{e^{n\rho}}{2\pi}\int_{-\pi}^{\pi}\exp\Big(f(\rho+it,\rho)+nt i\Big)dt.\end{equation}

We now use the saddle-point method to estimate the above integral. We choose $\rho$ to be the positive solution of the equation

(3·5) \begin{equation}n=-f_{x}(\rho,\rho)=\sum_{l=1}^{r}\sum_{m=1}^{\infty}\sum_{k=1}^{m}\frac{me^{-\rho(m+(c+ak+mbl))}}{1-e^{-\rho(m+(c+ak+mbl))}},\end{equation}

where $f_x$ denotes the partial derivative of f with respect to x. Similar notations will be used for other partial derivatives. Note that there is a unique positive solution $\rho=\rho(n,a,b,c)$ of Equation (3·5) since the function defined by the series is strictly decreasing as a function of $\rho$ , provided that a, b and c are small enough (it suffices for instance to assume that $|c|+|a|+r|b|<1$ ). Furthermore, we observe that $\rho\to0$ as $n\to \infty$ . The following lemma reveals the asymptotic dependence between n and $\rho$ .

Lemma 3·3. Let $\epsilon$ be a number in the interval $[0,1/2]$ and $\rho$ be the solution of Equation (3·5). Then we have

(3·6) \begin{equation}\frac{2r\zeta(3)}{(1+\epsilon)^3} \, \rho^{-3}+\mathcal{O}(\rho^{-2})\leq n \leq \frac{2r\zeta(3)}{(1-\epsilon)^3}\, \rho^{-3}+\mathcal{O}(\rho^{-2}),\end{equation}

as $n\to\infty$ , uniformly for $|c|+|a|+r|b|\leq \epsilon$ , where the implied constants in the $\mathcal{O}$ -terms are independent of $\epsilon.$

Proof. Recall from (3·5) that

\begin{equation*}n =-f_x(\rho,\rho)=-\sum_{l=1}^{r}\sum_{m=1}^{\infty}\sum_{k=1}^{m}m\Phi'\!\left(\rho m\left(1+\frac{c+ak+mbl}{m}\right)\right).\end{equation*}

Under the assumption that $|c|+|a|+r|b|\leq \epsilon$ , for any $m\geq 1$ , $1\leq k\leq m$ and $1\leq l\leq r$ , we have

\begin{equation*}|c+ak+mbl|\leq \left(|c|+|a|+r|b|\right) m\leq \epsilon m.\end{equation*}

Moreover, the function $\Phi'(x)$ is an increasing function. Therefore,

\begin{equation*}\Phi'((1-\epsilon)\rho m)\leq \Phi'\left(\rho m\left(1+\frac{c+ak+mbl}{m}\right)\right)\leq \Phi'( (1+\epsilon)\rho m).\end{equation*}

Multiplying by m and summing over $m\geq 1$ , $1\leq k\leq m$ and $1\leq l\leq r$ , we obtain

\begin{equation*}f_x((1-\epsilon)\rho,0)\leq f_x(\rho,\rho)\leq f_x((1+\epsilon)\rho,0).\end{equation*}

We can obtain asymptotic estimates of the lower and upper bounds as $\rho\to0^+$ . This can be done via Mellin transform. The reader can consult [Reference Flajolet, Gourdon and Dumas17] for a comprehensive survey on the Mellin transform method. The Mellin transform of $f_x((1-\epsilon)t,0)$ is

\begin{equation*}\int_{0}^{\infty} f_x((1-\epsilon)t,0) t^{s-1} dt = -r(1-\epsilon)^{-s}\zeta(s-2)\zeta(s)\Gamma(s),\end{equation*}

which has simple poles at $s=3$ and $s=1$ . The other singularities are precisely at the negative odd integers. Thus, we have

(3·7) \begin{align} f_x((1-\epsilon)\rho,0) & =-\frac{2r\zeta(3)}{(1-\epsilon)^{3}}\, \rho^{-3}\nonumber\\ & \quad +\frac{r}{12(1-\epsilon)}\, \rho^{-1}-\frac{r}{2\pi i }\int_{-i\infty}^{i\infty}\zeta(s-2)\zeta(s)\Gamma(s)((1-\epsilon)\rho)^{-s}ds.\end{align}

Since $|\zeta(it-2)\zeta(it)\Gamma(it)|$ decays exponentially fast as $t\to \pm\infty$ , the absolute value of the integral on the right-hand side is bounded by an absolute constant. The same argument works for the estimate of the upper bound $f_x((1+\epsilon)\rho,0)$ . This completes the proof of the lemma.

Next, we split the integral on the right-hand side of (3·4) into two parts as follows: let

\begin{equation*}\mathcal{I}_1=\frac{e^{n\rho}}{2\pi}\int_{-\rho^{C}}^{\rho^{C}}\exp\Big(f(\rho+it,\rho)+nti\Big)\,dt,\end{equation*}

where C is an absolute constant in the interval $(5/3, 2)$ , and let $\mathcal{I}_2=Q_{n}(e^{-a\rho},e^{-b\rho},e^{-c\rho})-\mathcal{I}_1$ .

Estimate of $\mathcal{I}_1$

For the rest of this section, we work under the condition of Lemma 3·3, that is $|c|+|a|+r|b|\leq \epsilon$ and $\epsilon\in [0,1/2]$ . For $-\rho^{C}\leq t\leq \rho^{C}$ , Equation (3·5) and a Taylor approximation of $f(\rho+it,\rho)$ give

\begin{equation*}f(\rho+it,\rho)+nit=f(\rho,\rho)-f_{xx}(\rho,\rho)\frac{t^2}{2}+\mathcal{O}\left(\rho^{3C}\max_{-\rho^{\rho}\leq \theta\leq \rho^{C} }\Big|f_{xxx}(\rho+i\theta,\rho)\Big|\right),\end{equation*}

where the implied constant in the error term is absolute. To estimate the error term, observe that

\begin{equation*}f_{xxx}(\tau,\rho)=-\sum_{l=1}^{r}\sum_{m=1}^{\infty}\sum_{k=1}^{m}\frac{m^3e^{-\tau m-\rho(c+ak+mbl)}(1+e^{-\tau m-\rho(c+ak+mbl)})}{(1-e^{-\tau m-\rho(c+ak+mbl)})^3}.\end{equation*}

For any real number $\theta$ and $\tau=\rho+i\theta$ we have

\begin{align*}|1+e^{-\tau m-\rho(c+ak+mbl)}|& \leq 1+e^{-\rho m-\rho(c+ak+mbl)}, \\|1-e^{-\tau m-\rho(c+ak+mbl)}| & \geq 1-e^{-\rho m-\rho(c+ak+mbl)}.\end{align*}

Hence $|f_{xxx}(\rho+i\theta,\rho)|$ is bounded above by $|f_{xxx}(\rho,\rho)|$ . We can estimate $|f_{xxx}(\rho,\rho)|$ as we did for $f_{x}(\rho,\rho)$ in the proof of Lemma 3·3. We obtain

\begin{equation*}|f_{xxx}(\rho+i\theta,\rho)|\leq |f_{xxx}(\rho,\rho)| =\mathcal{O}(\rho^{-5}),\end{equation*}

where the implied constant depends only on r. Therefore, for $\lvert t \rvert\leq \rho^{C}$ we have

\begin{equation*}f(\rho+it,\rho)+nit=f(\rho,\rho)-f_{xx}(\rho,\rho)\frac{t^2}{2}+\mathcal{O}(\rho^{3C-5}).\end{equation*}

Since we chose $C>5/3$ , we have $\rho^{3C-5}=o(1)$ . Thus

\begin{equation*}\mathcal{I}_1=\frac{e^{f(\rho,\rho)+n\rho}}{2\pi}\int_{-\rho^C}^{\rho^C}e^{-f_{xx}(\rho,\rho)t^2/2}\, dt\, \Big(1+\mathcal{O}(\rho^{3C-5})\Big).\end{equation*}

It remains to estimate the integral on the right-hand side as $\rho\to 0^+$ . Note that $f_{xx}(\rho,\rho)>0$ and $f_{xx}(\rho,\rho) \gg \rho^{-4} $ (again via Mellin transform as in Lemma 3·3), so we have

\begin{align*}\int_{-\rho^{C}}^{\rho^{C}}e^{-f_{xx}(\rho,\rho)t^2/2}\, dt& = \int_{-\infty}^{\infty}e^{-f_{xx}(\rho,\rho)t^2/2}\, dt-2\int_{\rho^{C}}^{\infty}e^{-f_{xx}(\rho,\rho)t^2/2}\, dt\\& =\sqrt{\frac{2\pi}{f_{xx}(\rho,\rho)}}+\mathcal{O}\left(\int_{\rho^{C}}^{\infty}e^{-\rho^{C}f_{xx}(\rho,\rho)t/2}\, dt\right)\\& =\sqrt{\frac{2\pi}{f_{xx}(\rho,\rho)}}+\mathcal{O}\left(\rho^{4-c}e^{-A\rho^{2C-4}}\right),\end{align*}

where $A>0$ and the hidden constants in the error terms above depend only on r. Thus, since we chose $C<2$ , the term $\rho^{4-c}e^{-A\rho^{2C-4}}$ tends to zero faster than any power of $\rho$ as $\rho\to0^{+}$ . Hence, we obtain an estimate for $\mathcal{I}_1$

(3·8) \begin{equation}\mathcal{I}_1=\frac{e^{\,f(\rho,\rho)+n\rho}}{\sqrt{2\pi f_{xx}(\rho,\rho)}}\Big(1+\mathcal{O}(\rho^{3C-5})\Big)\ \ \text{as}\ \ \rho\to 0^{+}.\end{equation}

This estimate holds uniformly for $|c|+|a|+r|b|\leq \epsilon$ and $\epsilon\in [0,1/2]$ .

Estimate of $\mathcal{I}_2$

We will prove that $|\mathcal I_2|$ is much smaller than $|\mathcal I_1|$ . To this end, we assume that $\rho^{C}< t \leq \pi$ , where C is as before. We have

\begin{align*} \text{Re} (f(\rho+it,\rho))-f(\rho,\rho)& \,\,=\,\,-\sum_{l=1}^r\sum_{m=1}^{\infty}\sum_{k=1}^{m}\sum_{j=1}^{\infty}j^{-1}e^{-j\rho(m+(c+ak+mbl))}(1-\cos\!(mt))\\& \,\,\leq\,\, -\sum_{l=1}^r\sum_{m=1}^{\infty}\sum_{k=1}^{m}e^{-\rho(m+(c+ak+mbl))}\left(1-\cos\!(mt)\right)\\& \,\,\leq\,\, -r\sum_{m=1}^{\infty}me^{-\rho(1+\epsilon)m}(1-\cos\!(mt)).\end{align*}

Moreover,

\begin{equation*}\sum_{m=1}^{\infty}me^{-\rho(1+\epsilon)m}(1-\cos\!(mt)) =\frac{e^{\rho(1+\epsilon)}}{(e^{\rho(1+\epsilon)}-1)^2}-\text{Re}\left( \frac{e^{\rho(1+\epsilon)+it}}{(e^{\rho(1+\epsilon)+it}-1)^2}\right).\end{equation*}

A lower estimate of the same term can be found in the proof of [Reference Luca and Ralaivaosaona21, lemma 5]. By the same argument as the one given in loc. cit., but with $\lvert t \rvert\geq \rho^{C}$ , we get

\begin{equation*}\sum_{m=1}^{\infty}me^{-\rho(1+\epsilon)m}(1-\cos\!(mt)) \gg \left(\rho(1+\epsilon)\right)^{2C-4} \ \ \text{as}\ \ \rho\to0^+,\end{equation*}

where the implied constant is independent of $\epsilon.$

Noting that $2C-4<0$ , we deduce that $\exp\!\left( \text{Re} (f(\rho+it,\rho))-f(\rho,\rho)\right)$ tends to zero faster than any power of $\rho$ as $\rho\to0^+$ . Thus, by (3·8), we find

\begin{equation*}\frac{|\mathcal{I}_2|}{|\mathcal{I}_1|}\ll \sqrt{f_{xx}(\rho,\rho)} \int_{\rho^C}^{\pi}\exp\left( \text{Re} (f(\rho+it,\rho))-f(\rho,\rho)\right)dt,\end{equation*}

which tends to zero faster than any power of $\rho$ as $\rho\to0^+$ . Recalling that $Q_{n}(e^{-a\rho},e^{-b\rho},e^{-c\rho})=\mathcal{I}_1+\mathcal{I}_2$ , we finally obtain

(3·9) \begin{equation} Q_{n}(e^{-a\rho},e^{-b\rho},e^{-c\rho})=\frac{e^{f(\rho,\rho)+n\rho}}{\sqrt{2\pi f_{xx}(\rho,\rho)}}\Big(1+\mathcal{O}(\rho^{3C-5})\Big)\ \ \text{as}\ \ n\to \infty,\end{equation}

uniformly for $|c|+|a|+r|b|\leq \epsilon$ and $\epsilon\in [0,1/2]$ , where $\rho$ is the solution of Equation (3·5).

3·3·2. Perturbation

Here we set $u=e^{\eta\alpha}$ , $v=e^{\eta\beta}$ , and $w=e^{\eta\gamma}$ where $(\alpha,\beta,\gamma)$ is fixed and $\eta$ can vary in a small open interval containing zero. Hence, Equation (3·2) becomes

(3·10) \begin{equation} \frac{Q_n(u,v,w)}{Q_n(1,1,1)}=\mathbb{E}\left(e^{\eta(\alpha X_n+\beta Y_n+\gamma Z_n)}\right).\end{equation}

The right-hand side is the moment generating function of the random variable $\alpha X_n+\beta Y_n+\gamma Z_n$ . From now on, let $\rho_0$ be the unique positive number such that $n=-f_x(\rho_0,0).$ Then, by Lemma 3·3 (with $\epsilon= 0$ ) we have $n\sim 2r\zeta(3)\rho_0^{-3}.$ Moreover, if we write $u=e^{-a \rho}$ , $v=e^{-b \rho}$ , and $w=e^{-\gamma\rho}$ for $\rho>0$ as before, then we have $a=-\alpha\eta \rho^{-1}$ , $b=-\beta\eta \rho^{-1}$ , and $c=-\gamma\eta \rho^{-1}$ . This implies that

\begin{equation*}|c|+|a|+r|b|=(|\gamma|+|\alpha|+r|\beta|)\eta\rho^{-1}.\end{equation*}

Now, if we choose $\eta$ and $\rho$ in such a way that $\eta\rho^{-1}=o(1)$ and $\rho\to 0$ as $n\to \infty$ , then by Lemma 3·3 (with $\epsilon\to 0$ ), we get

\begin{equation*}-f_x(\rho,\rho)\sim 2r\zeta(3)\rho^{-3}.\end{equation*}

Observe that it is possible to choose such $\eta$ and $\rho>0$ that satisfy $\eta=o(\rho_0)$ and $\rho=o(\rho_0)$ . In this case, the above asymptotic formula implies that $-f_x(\rho,\rho)>-f_x(\rho_0,0)=n$ for large enough n. Similarly, we can also choose $\eta$ and $\rho>0$ such that $\eta=o(\rho_0)$ , $\rho\to 0$ , $\eta\rho^{-1}=o(1)$ , and ${\rho}/{\rho_0}\to \infty$ . This time, the asymptotic formula gives $-f_x(\rho,\rho)<n$ . Hence, for $\eta=o(\rho_0)$ and n large enough, the equation $n=-f_x(\rho,\rho)$ has a unique solution, which we denote by $\rho(\eta)$ . Furthermore, it satisfies $\rho(\eta)\to 0$ and $\eta\rho(\eta)^{-1}=o(1)$ as $n\to \infty$ . Therefore, we also have

\begin{equation*}n\sim 2r\zeta(3)\rho(\eta)^{-3}.\end{equation*}

The latter and the asymptotic estimate $n\sim 2r\zeta(3)\rho_0^{-3}$ yield $\rho(\eta)\sim \rho_0$ as $n\to \infty$ whenever $\eta=o(\rho_0)$ .

From this point onward, we assume that $\eta=o(\rho_0).$ Since $\rho(\eta)$ and $\rho_0$ are asymptotically equivalent as $n\to \infty$ , so are $f_{xx}(\rho(\eta), \rho(\eta))$ and $f_{xx}(\rho_0, 0).$ Thus, by (3·9), we have

(3·11) \begin{equation} \frac{Q_n(u,v,w)}{Q_n(1,1,1)}\sim \exp\Big(f(\rho(\eta),\rho(\eta))-f(\rho_0,0)+n(\rho(\eta)-\rho_0)\Big).\end{equation}

We want to obtain a precise asymptotic estimate of the exponent of the right-hand side that holds uniformly for $\eta=o(\rho_0)$ . We will use a Taylor approximation of the function $f(\rho(\eta),\rho(\eta))$ when $\eta$ is near 0 (noting that the parameters a, b, and c are themselves functions of $\eta$ ). So to highlight the variable $\eta$ , we define

\begin{align*} g(x,y) & =\log F(e^{y\alpha},e^{y\beta},e^{y\gamma},e^{-x})\\ &=-\sum_{l=1}^{r}\sum_{m=1}^{\infty}\sum_{k=1}^{m}\log\left(1-e^{y(\gamma+\alpha k+m\beta l)-x m}\right)\\ &=\sum_{l=1}^{r}\sum_{m=1}^{\infty}\sum_{k=1}^{m}\Phi(xm-y(\gamma+\alpha k+m\beta l)).\end{align*}

This is essentially the same as the function f(x, y) (if (a, b, c) is replaced by $(\!-\alpha, -\beta, -\gamma)$ ). However, in our case we have $a=-\alpha\eta \rho(\eta)^{-1}$ , $b=-\beta\eta \rho(\eta)^{-1}$ , and $c=-\gamma\eta \rho(\eta)^{-1}$ . Hence, the meanings of the first and second variables will be different. For instance, we have the equation

(3·12) \begin{equation} g(\rho(\eta),\eta)=f(\rho(\eta),\rho(\eta))\ \ \text{and}\ \ g(\rho_0,0)=f(\rho_0,0).\end{equation}

Similarly, the saddle-point equation $n=-f_x(\rho(\eta),\rho(\eta))$ becomes $n=-g_x(\rho(\eta),\eta).$ First, we apply implicit differentiation (with respect to $\eta$ ) to the latter equation, then by the mean value theorem, we get

\begin{equation*}\rho(\eta)-\rho_0= -\eta \, \frac{g_{xy}(\rho(\theta),\theta)}{g_{xx}(\rho(\theta),\theta)},\end{equation*}

for some real number $\theta$ between 0 and $\eta.$ We can estimate $g_{xy}(\rho(\theta),\theta)$ and $g_{xx}(\rho(\theta),\theta)$ via Mellin transform in the same way as in the proof of Lemma 3·3, but we need the Mellin transforms of the functions $g_{xx}(t,0)$ and $g_{xy}(t,0)$ . The Mellin transform of $g_{xx}(t,0)$ is $r\zeta(s-3)\zeta(s-1)\Gamma(s)$ . To determine the Mellin transform of $g_{x,y}(t,0)$ , first we write

\begin{align*} g_{x,y}(t,0) & \,\,=\,\,-\sum_{l=1}^{r}\sum_{m=1}^{\infty}\sum_{k=1}^{m}m(\gamma+\alpha k+m\beta l)\Phi''(tm)\\ & \,\,=\,\, -\sum_{m=1}^{\infty}m\!\left(rm\gamma+r\alpha\frac{m^2+m}{2}+m^2\beta\frac{r^2+r}{2}\right)\Phi''(tm)\\ & \,\,=\,\, -\frac{r}{2}\sum_{m=1}^{\infty}\left((\alpha+(r+1)\beta) m^3+(\alpha+2\gamma)m^2\right)\Phi''(tm).\end{align*}

Since the Mellin transform of $\Phi''(t)$ is $\zeta(s-1)\Gamma(s)$ , the Mellin transform of $g_{x,y}(t,0)$ is

\begin{equation*}-\frac{r}{2}\left((\alpha+(r+1)\beta)\zeta(s-3) +(\alpha+2\gamma)\zeta(s-2)\right)\zeta(s-1)\Gamma(s).\end{equation*}

We deduce the following asymptotic formulas as $\rho\to0^{+}$ :

(3·13) \begin{equation} g_{x,x}(\rho,0) =6r\zeta(3)\rho^{-4}+\mathcal{O}(\rho^{-2}),\end{equation}

(3·14) \begin{equation} g_{x,y}(\rho,0) =-3r\zeta(3)(\alpha+(r+1)\beta)\rho^{-4}-r\zeta(2)(\alpha+2\gamma)\rho^{-3}+\mathcal{O}(\rho^{-2}).\end{equation}

The fact that $\rho(\theta)\sim \rho_0$ (since $|\theta|\leq |\eta|=o(\rho_0)$ ) and the argument in the proof of Lemma 3·3 (with $\epsilon\to 0$ ) imply

(3.15) \begin{equation} \rho(\eta)-\rho_0\sim -\eta \, \frac{g_{xy}(\rho_0,0)}{g_{xx}(\rho_0,0)}=\mathcal{O}(|\eta|)=o(\rho_0).\end{equation}

Similarly, we have the following estimate for the third partial derivatives

\begin{equation*}\frac{\partial^k}{\partial x^k}\frac{\partial^l}{\partial y^l} g (x,y)\Big|_{(x,y)=(\rho(\theta),\theta)}=\mathcal{O}\Big(\rho_0^{-5}\Big),\end{equation*}

uniformly for $\theta=o(\rho_0)$ , and for any nonnegative integers k and l such that $k+l=3$ . This shows that if $\eta=o(\rho_0)$ , then we have the Taylor approximation

$$\matrix{ {g(\rho (\eta ),\eta ) = g({\rho _0},0) + {g_y}({\rho _0},0)\eta + {g_x}({\rho _0},0)(\rho (\eta ) - {\rho _0})} \hfill \cr { + {1 \over 2}({g_{yy}}({\rho _0},0){\mkern 1mu} {\eta ^2} + 2{g_{xy}}({\rho _0},0){\mkern 1mu} \eta (\rho (\eta ) - {\rho _0}) + {g_{xx}}({\rho _0},0){\mkern 1mu} {{(\rho (\eta ) - {\rho _0})}^2}) + {\cal O}(|\eta {|^3}\rho _0^{ - 5}),} \hfill \cr } $$

where the implied constant in the error term is independent of n. Using (3·15) and saddle-point equation $n=-g_x(\rho_0,0)$ , we deduce that

$$\matrix{ {g(\rho (\eta ),\eta ) - g({\rho _0},0) + n(\rho (\eta ) - {\rho _0}) = } \hfill \cr {\quad \quad \quad \quad \quad \quad \quad {g_y}({\rho _0},0)\eta + \left( {{{{g_{yy}}({\rho _0},0){g_{xx}}({\rho _0},0) - {{({g_{xy}}({\rho _0},0))}^2}} \over {{g_{xx}}({\rho _0},0)}}} \right){{{\eta ^2}} \over 2} + {\cal O}(|\eta {|^3}\rho _0^{ - 5}).} \hfill \cr } $$

Let us define

(3·16) \begin{equation} \sigma_n^2=\frac{g_{yy}(\rho_0,0)g_{xx}(\rho_0,0)-(g_{xy}(\rho_0,0))^2}{g_{xx}(\rho_0,0)} \, \text{ and }\, \mu_n=g_y(\rho_0,0),\end{equation}

and we choose $\eta={t}/{\sigma_n}$ where t is a fixed real number. Then our estimate (3·11) becomes

(3·17) \begin{equation} \frac{Q_n(u,v,w)}{Q_n(1,1,1)}\sim \exp\left(\frac{\mu_n}{\sigma_n}\ t+\frac{t^2}{2}+\mathcal{O}(\sigma_n^{-3}\rho_0^{-5})\right),\end{equation}

as $n\to\infty$ (this is valid as long as $\eta={t}/{\sigma_n}=o(\rho_0)$ , but we do not even know at this stage whether $\sigma_n^2$ is positive). Hence, let us estimate $\sigma_n^2$ .

Assume first that $(\alpha,\beta)\neq (0,0)$ . Once again, by the Mellin transform technique, we have

(3·18) \begin{equation} g_{yy}(\rho_0,0) \sim r\left(2\alpha^2+3(r+1)\alpha\beta+(r+1)(2r+1)\beta^2\right)\zeta(3)\rho_0^{-4}.\end{equation}

Putting the estimates (3·13), (3·14) and (3·18) into the formula for $\sigma_n^2$ in (3·16), we have

(3·19) \begin{equation} \sigma_n^2\sim \left(\frac{r\zeta(3)}{2}\, \alpha^2+\frac{r(r^2-1)\zeta(3)}{2}\, \beta^2\right)\rho_0^{-4}.\end{equation}

Hence, $\eta=\mathcal{O}(\rho_0^2)$ and the estimate (3·17) becomes

\begin{equation*}\frac{Q_n(u,v,w)}{Q_n(1,1,1)}\sim \exp\left(\frac{\mu_n}{\sigma_n}\ t+\frac{t^2}{2}+\mathcal{O}(\rho_0)\right).\end{equation*}

Therefore, if $(\alpha,\beta)\neq (0,0)$ and t is any fixed real number, then the latter identity together with (3·10) yield

\begin{equation*}e^{-\mu_nt\sigma_n^{-1}}\mathbb{E}\left(e^{t\, \sigma_n^{-1}(\alpha X_n+\beta Y_n+\gamma Z_n)}\right)\sim e^{t^2/2},\end{equation*}

as $n\to \infty$ . This means, by Curtiss’ theorem [Reference Curtiss10], that

\begin{equation*}\frac{\alpha X_n+\beta Y_n+\gamma Z_n-\mu_n}{\sigma_n}\xrightarrow{\text{d}}{\to} \mathcal{N}(0,1) \ \ \text{as}\ \ n\to\infty.\end{equation*}

To obtain the asymptotic formulas for $\sigma_n^2$ and $\mu_n$ , recall from the saddle-point equation and (3·7) (with $\epsilon=0$ ) that $n= 2r\zeta(3)\rho_0^{-3}-{1}/{12}r\rho_0^{-1}+\mathcal{O}(1)$ . Inverting this yields

(3·20) \begin{equation} \rho_0^{-1}=\frac{n^{1/3}}{(2r\zeta(3))^{1/3}}+\mathcal{O}(n^{-1/3}).\end{equation}

So we first estimate $\sigma_n^2$ and $\mu_n$ in terms of $\rho_0$ , then use the above to get the asymptotic formulas in terms of n. Such an estimate for $\sigma_n^2$ is already given in (3·19). The estimate of $\mu_n$ can be obtained easily from the Mellin transform of $g_y(t,0)$ , which is

\begin{equation*}\frac{r}{2}\left(\left(\alpha+(r+1)\beta\right)\zeta(s-2)+\left(\alpha+2\gamma\right)\zeta(s-1)\right)\zeta(s)\Gamma(s).\end{equation*}

By a straightforward calculation, we have

\begin{equation*}\sigma_n^2\sim \frac{\alpha^2+(r^2-1)\beta^2}{2^{7/3}(r\zeta(3))^{1/3}}\, n^{4/3}\kern-1.5pt\, \text{ and }\,\kern-1.5pt \mu_n= \left(\!\frac{1}{2}\alpha+\frac{r+1}{2} \beta\!\right)\!n+\frac{r^{1/3}\zeta(2)(\alpha+2\gamma)}{2^{5/3}(\zeta(3))^{2/3}}\, n^{2/3}+\mathcal{O}(n^{1/3}).\end{equation*}

If we now assume that $(\alpha,\beta)=(0,0)$ but $\gamma\neq 0$ , then the Mellin transform of $g_{yy}(t,0)$ is $r\gamma^2\zeta(s-1)^2\Gamma(s)$ whose dominant singularity is a double pole at $s=2$ . This leads to the asymptotic formula $g_{yy}(\rho_0,0)= r\gamma^2\rho_0^{-2}\log\left(\rho_0^{-1}\right)+\mathcal{O}(\rho_0^{-2}).$ Hence, the formula in (3·16) gives

(3·21) \begin{equation}\sigma_n^2=r\gamma^2\rho_0^{-2}\log\left(\rho_0^{-1}\right)+\mathcal{O}(\rho_0^{-2}).\end{equation}

Thus, for a fixed real number t, we have

\begin{equation*}\eta=\frac{t}{\sigma_n}=\mathcal{O}(\rho_0|\log \rho_0|^{-1/2}).\end{equation*}

So we still have our desired condition that $\eta=o(\rho_0)$ . Moreover, applying (3·13) and (3·14) with $(\alpha,\beta)=(0,0)$ , the estimate (3·15) becomes

\begin{equation*}\rho(\eta)-\rho_0\sim -\eta \, \frac{g_{xy}(\rho_0,0)}{g_{xx}(\rho_0,0)}=\mathcal{O}(|\eta| \rho_0)=\mathcal{O}(\rho_0^2|\log \rho_0|^{-1/2}).\end{equation*}

On the other hand, for any $\theta=o(\rho_0)$ , we have the following:

\begin{align*} g_{xxx}(\rho(\theta),\theta)&\,\,=\,\,\mathcal{O}(\rho_0^{-5}), & g_{xxy}(\rho(\theta),\theta)\,\,=\,\,\mathcal{O}(\rho_0^{-4}),\\ g_{xyy}(\rho(\theta),\theta)&\,\,=\,\,\mathcal{O}(\rho_0^{-3}|\log \rho_0|), & g_{yyy}(\rho(\theta),\theta)\,\,=\,\,\mathcal{O}(\rho_0^{-3}).\end{align*}

Therefore, (3·17) becomes

\begin{equation*} \frac{Q_n(u,v,w)}{Q_n(1,1,1)}\sim \exp\left(\frac{\mu_n}{\sigma_n}\ t+\frac{t^2}{2}+\mathcal{O}\left((\log n)^{-3/2}\right)\right). \end{equation*}

Just as in the previous case, this is enough to prove the central limit theorem. The asymptotic formula for the variance in terms of n can be obtained from (3·21) using (3·20). The proof of Proposition 3·1 is complete.