1. Introduction
Given an integer $n \geqslant 1$ , it is natural to study the distribution of its divisors over the interval [1,n] (in logarithmic scale). Let d be a random integer chosen uniformly from the divisors of n. Then $D_n\;:\!=\;{\log d}/{\log n}$ is a random variable taking values in $[0,1].$ While one can show that the sequence of random variables $\{D_n\}_{n=1}^{\infty}$ does not converge in distribution, Deshouillers, Dress and Tenenbaum [ Reference Deshouillers, Dress and Tenenbaum5 ] proved the mean of the corresponding distribution functions converges to that of the arcsine law. More precisely, uniformly for $u \in [0,1],$ we have
where
is the distribution function of $D_n$ and the error term here is optimal (see also [ Reference Tenenbaum20 , chapter 6·2]).
Recently, Nyandwi and Smati [ Reference Nyandwi and Smati16 ] studied the distribution of pairs of divisors of a given integer on average. Similarly, they also proved the mean of the corresponding distribution functions converges to that of the beta two-dimensional law uniformly together with the optimal rate of convergence.
Our main aim here is to generalise their work to higher dimensions, which they claim is very technical following the usual approach (see [ Reference Nyandwi and Smati17 , p. 2]). Fix $k \geqslant 2.$ Given an integer $n \geqslant 1,$ let $(d_1,\ldots,d_k)$ be a random k-tuple chosen uniformly from the set of all possible factorisation $\{(m_1,\ldots,m_k) \in \mathbb{N}^k \, : \, n=m_1\cdots m_k\}.$ Then $\boldsymbol{D}_n=(D_n^{(1)},\ldots,D_n^{(k)})\;:\!=\;\left({\log d_1}/{\log n},\ldots,{\log d_k}/{\log n}\right)$ is a multivariate random variable taking values in $[0,1]^k.$ Similarly, we are interested in the mean
where
is the distribution function of $\boldsymbol{D}_n.$
Note that since $n=d_1\cdots d_k,$ the multivariate random variable $\boldsymbol{D}_n$ must satisfy
and so it actually takes values in the $(k-1)$ -dimensional probability simplex. We now turn to the Dirichlet distribution, which is the most natural candidate of modeling such distribution.
Definition 1·1. Let $k \geqslant 2.$ The Dirichlet distribution of dimension k with parameters $\alpha_1,\ldots,\alpha_k>0$ is denoted by $\mathrm{Dir}\left(\alpha_1,\ldots,\alpha_k\right), $ which is defined on the $(k-1)$ -dimensional probability simplex
having density
For instance, when $k=2,$ Dirichlet distribution reduces to beta distribution $\mathrm{Beta}\left(\alpha, \beta\right)$ with parameters $\alpha, \beta$ . In particular, $\mathrm{Beta}\left({1}/{2}, {1}/{2}\right)$ is the arcsine distribution.
As we will see, factorisation of integers into k parts follows the Dirichlet distribution $\mathrm{Dir}\left({1}/{k},\ldots,{1}/{k}\right).$ Since for each i the parameter $\alpha_i={1}/{k}$ is less than 1, the density $f_{\boldsymbol{\alpha}}(t_1,\ldots,t_k)$ blows up most rapidly at the k vertices of the probability simplex. Therefore, our intuition that a typical factorisation of integers into k parts consists of one large factor and $k-1$ small factors is justified quantitatively.
By definition, for $u_1,\ldots,u_{k-1} \geqslant 0$ satisfying $u_1+\cdots+u_{k-1}\leqslant 1,$ the distribution function of $\mathrm{Dir}\left(\alpha_1,\ldots,\alpha_k \right)$ is given by
From now on until Section 6, we shall fix $\boldsymbol{\alpha}=\left({1}/{k},\ldots,{1}/{k}\right)$ and omit the subscript.
The main results are stated as follows.
Theorem 1·1. Let $k\geqslant 2$ be a fixed integer. Then uniformly for $x \geqslant 2$ and $u_1,\ldots,u_{k-1} \geqslant 0$ satisfying $u_1+\cdots+u_{k-1}\leqslant 1,$ we have
The error term here is optimal if full uniformity in $u_1,\ldots,u_{k-1}$ is required. Indeed, if we choose $u_1=\cdots=u_{k-2}=\frac{1}{k}, u_{k-1}=0,$ then one can show that the left-hand side of (1·1) is of order $(\!\log x)^{-\frac{1}{k}}$ using [ Reference Deshouillers, Dress and Tenenbaum5 , théorème T] followed by partial summation.
Remark 1·1. Instead of using the logarithmic scale, one may also study localised factorisation of integers, say for instance the quantity
which was discussed in [ Reference Koukoulopoulos13 ].
Note that Theorem 1·1 implies that for any axis-parallel rectangle $R \subseteq \Delta^{k-1},$ we have
Since every Borel subset of the simplex can be approximated by finite unions of such rectangles, the following corollary is an immediate consequence of Theorem 1·1.
Corollary 1·1. Let $k \geqslant 2$ be a fixed integer. For $x \geqslant 1,$ let n be a random integer chosen uniformly from [1,x] and $(d_1,\ldots,d_k)$ be a random k-tuple chosen uniformly from the set of all possible factorisation $\{(m_1,\ldots,m_k) \in \mathbb{N}^k\, : \, n=m_1\cdots m_k\}.$ Then as $x \to \infty,$ we have the convergence in distribution
It is a general phenomenon that the “anatomy” of polynomials or permutations is essentially the same as that of integers (see [ Reference Granville9, Reference Granville and Granville10 ]), and the main theorem here is no exception. In the realm of polynomials, the following theorem serves as the counterpart to Theorem 1·1.
Theorem 1·2. Let $k\geqslant 2$ be a fixed integer and q be a fixed prime power. Then uniformly for $n \geqslant 1$ and $u_1,\ldots,u_{k-1}\geqslant 0$ satisfying $u_1+\cdots+u_{k-1}\leqslant 1$ , we have
where the notations are defined in Section 2.
Corollary 1·2. Let $k \geqslant 2$ be a fixed integer and q be a fixed prime power. For $n \geqslant 1,$ let F be a random polynomial chosen uniformly from $\mathcal{M}_q(n)$ and $(D_1,\ldots,D_k)$ be a random k-tuple chosen uniformly from the set of all possible factorisation $\{(G_1,\ldots,G_k) \in \mathcal{M}_q^k\, : \, F=G_1\cdots G_k\}.$ Then as $n \to \infty,$ we have the convergence in distribution
Similarly, in the realm of permutations, the following theorem serves as the counterpart to Theorem 1·1.
Theorem 1·3. Let $k\geqslant 2$ be a fixed integer. Then uniformly for $n \geqslant 1$ and $u_1,\ldots,u_{k-1}\geqslant 0$ satisfying $u_1+\cdots+u_{k-1}\leqslant 1$ , we have
where the notations are defined in Section 2.
Corollary 1·3. Let $k \geqslant 2$ be a fixed integer. For $n \geqslant 1,$ let $\sigma$ be a random permutation chosen uniformly from $S_n$ and $(A_1,\ldots, A_k)$ be a random k-tuple chosen uniformly from the set of all possible $\sigma$ -invariant decomposition $\{(B_1,\ldots, B_k) \, : \, [n]=B_1\sqcup \cdots \sqcup B_k, \sigma(B_i)=B_i \text{ for }i=1,\ldots, k\}.$ Then as $n \to \infty,$ we have the convergence in distribution
In Section 7, we model the Dirichlet distribution with arbitrary parameters by assigning probability weights which are not necessarily uniform to each integer and to each factorisation. Then, as we will see, most of the results in the literature about the distribution of divisors in logarithmic scale are direct consequences of Theorem 7·1, which is a generalisation of Theorem 1·1.
2. Notation
Throughout the paper, we shall adopt the following list of notation:
-
(a) we say $f(x)=O(g(x))$ or $f(x) \ll g(x)$ if there exists a constant $C>0$ which might depend on $k,q, \boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{c}, \boldsymbol{\delta}$ such that $|f(x)| \leqslant C \cdot g(x)$ whenever $x >x_0$ for some $x_0>0;$
-
(b) $[n]\;:\!=\;\{1,2,\ldots, n\};$
-
(c) $\tau_k(n)\;:\!=\;|\{(d_1,\ldots, d_k) \in \mathbb{N}^k \, : \, n=d_1\cdots d_{k} \} |$ and $\tau(n)\;:\!=\;\tau_2(n);$
-
(d) $\mathcal{M}_q\;:\!=\;\{F \in \mathbb{F}_q[x] \, : \, F \text{ is monic}\};$
-
(e) $\mathcal{M}_q(n)\;:\!=\;\{F \in \mathcal{M}_q \, : \, \deg{F}=n\};$
-
(f) $\tau_k(F)\;:\!=\;|\{(D_1,\ldots, D_k) \in \mathcal{M}_q^k \, : \, F=D_1\cdots D_{k} \} |;$
-
(g) $S_n$ denotes the group of permutations on $[n];$
-
(h) $c(\sigma)$ denotes the number of disjoint cycles of the permutation $\sigma;$
-
(i) $\tau_{\alpha}(\sigma)\;:\!=\;{\alpha}^{c(\sigma)};$
-
(j) $\left[{n \atop k}\right]\;:\!=\;|\{\sigma \in S_n \, : \, c(\sigma)=k\}|$ denote (unsigned) Stirling numbers of the first kind.
3. Properties of $\mathcal{D}(s_1,\ldots, s_k)$
Both [ Reference Deshouillers, Dress and Tenenbaum5, Reference Nyandwi and Smati16 ] deal with the divisors one by one using [ Reference Deshouillers, Dress and Tenenbaum5 , théorème T] followed by partial summation. However, as k gets larger, it is increasingly laborious to achieve full uniformity as well as the optimal rate of convergence, especially when one of the u’s is small or $u_1+\cdots+u_{k-1}$ is close to 1. Instead, we would like to apply Mellin’s inversion formula to (the second derivative of) the multiple Dirichlet series $\mathcal{D}(s_1,\ldots, s_k)$ defined below, which allows a more symmetric approach to the problem so that all the divisors can be handled simultaneously. We first establish a few properties of the multiple Dirichlet series that are essential to the proof of Theorem 1·1.
Lemma 3·1. Let $\mathcal{D}(s_1,\ldots, s_k)$ denote the multiple Dirichlet series
Then $\mathcal{D}(s_1,\ldots, s_k)$ converges absolutely in the domain
and uniformly on any compact subset of $\Omega.$ In particular, $\mathcal{D}(s_1,\dots,s_k)$ is an analytic function of k variables in $\Omega.$
Proof. Let $\sigma_j\;:\!=\;\mathrm{Re}(s_j)$ for $j=1,\ldots,k.$ Then, since
the lemma follows.
Lemma 3·2. The multiple Dirichlet series $D(s_1,\dots,s_k)$ can be expressed as the Euler product
in the domain $\Omega$ defined above.
Proof. Let $y \geqslant 2$ and $\sigma_j\;:\!=\;\mathrm{Re}(s_j)$ for $j=1,\dots,k.$ Then, since
the finite product
is well-defined.
Let $S(y)\;:\!=\;\{n\geqslant 1 \,: \, p|n \text{ implies } p \leqslant y\}$ be the set of y-smooth numbers. Then, since
we have
The lemma follows by letting $y \to \infty.$
Lemma 3·3. For $j=1,\ldots,k,$ let $R_j \subseteq \left\{ s_j \in \mathbb{C} \,:\, \mathrm{Re}(s_j) > {3}/{4}, \, |\mathrm{Im}(s_j)| > {1}/{4} \right\}$ be a zero-free region for $\zeta(s_j).$ Then the multiple Dirichlet series $\mathcal{D}(s_1,\dots,s_k)$ can be continued analytically to the domain $\prod_{j=1}^k R_j.$ Moreover, we have the bound
Proof. Let $(s_1,\dots,s_k)\in \mathbb{C}^k$ with $\sigma_j\;:\!=\;\mathrm{Re}(s_j)>1$ for $j=1,\dots,k.$ Then by Lemma 3·2 we have the Euler product expression
where the kth root is understood as its principal branch.
For $j=1,\ldots,k,$ expanding the kth root as
we find that the factors of the Euler product are $1+O\left(\sum_{i=1}^{k}\sum_{j=1}^{k} p^{-(\sigma_i+\sigma_j)} \right) $ by Taylor’s theorem. Therefore, the function
can be continued analytically to the domain where $\mathrm{Re}(s_j) > {3}/{4}$ for $j=1,\dots,k,$ in which it is uniformly bounded.
On the other hand, for $(s_1,\dots,s_k) \in \prod_{j=1}^k R_j,$ we can express $\mathcal{D}(s_1,\dots,s_k)$ as
and so the lemma follows.
Lemma 3·4. In the open hypercube
we have the estimate
Proof. By (3·2) and the fact that
we have the power series representation
in Q for some constants $a_{i_1,\dots,i_k} \in \mathbb{C}.$ It follows that
where by (3·3) and Lemma 3·2, the leading coefficient
4. Proof of Theorem 1·1
We begin with writing
where the main term is
and the error term is
Let us first bound the error term (4·2). For $j=1,\dots,k-1$ with $u_j \neq 0,$ we write $n=d_jm$ for some integer m. Then $d_j>n^{u_j}$ implies $m<d_j^{(1-u_j)/u_j},$ and the number of ways of obtaining m as a product $d_1\cdots d_{j-1}d_{j+1}\cdots d_k$ is bounded by $\tau_{k-1}(m).$ It follows that
If $u_{j}\leqslant {1}/{2},$ then using [ Reference Koukoulopoulos14 , theorem 14·2], this is bounded by
Otherwise, it follows from the simple observation
that the expression (4·3) is bounded by
again using [ Reference Koukoulopoulos14 , theorem 14·2].
Now we are left with the main term (4·1). In order to apply Mellin’s inversion formula, we follow the treatment in [ Reference Granville and Koukoulopoulos11 ] and [ Reference Koukoulopoulos14 , chapter 13].
Lemma 4·1. Let $T \geqslant 1.$ Let $\phi, \psi \colon [0, \infty) \to \mathbb{R}$ be smooth functions supported on [0,1] and $\left[0, 1+{1}/{T}\right]$ respectively with
and
Moreover, for each integer $j\geqslant 0$ , their derivatives satisfy the growth condition $\phi^{(j)}(y), \psi^{(j)}(y) \ll_j T^j$ uniformly for $y\geqslant 0.$ Let $\Phi(s), \Psi(s)$ be the Mellin transform of $\phi(y), \psi(y)$ respectively for $1 \leqslant \mathrm{Re}(s) \leqslant 2$ , i.e.
and
Then we have the estimates
and
for $j\geqslant 1.$
Proof. See [ Reference Granville and Koukoulopoulos11 , theorem 4].
We need the following version of Hankel’s lemma to extract the main contribution from the multidimensional contour integral in the proof of Lemma 4·3.
Lemma 4·2. Let $x>1, \sigma>1$ and $\mathrm{Re}(\alpha)>1.$ Then we have
Proof. See [ Reference Koukoulopoulos14 , lemma 13·1].
We now prove the main lemma.
Lemma 4·3. Let $x_1,\dots,x_k \geqslant e$ and $S(x_1,\dots,x_k)$ denote the weighted sum
Then we have
with
As in [ Reference Granville and Koukoulopoulos11 ] and [ Reference Koukoulopoulos14 , chapter 13], we introduce powers of logarithms to ensure that the major contribution to the multiple Perron integral below comes from $s_1,\ldots,s_k \approx 1.$ Later on, they will be removed by partial summation.
Proof. The proof consists of four steps: Mellin inversion, localisation, approximation and completion. For $j=1,\ldots,k$ , let $T_j=2(\!\log x_j)^{2}$ and $\phi_j, \psi_j$ be any smooth functions coincide with $\phi, \psi$ respectively from Lemma 4·1. Then the weighted sum $S(x_1,\ldots,x_k)$ is bounded between
and
To avoid repetitions, we only establish the upper bound here. Applying Mellin’s inversion formula, the expression (4·7) becomes
Then, by Lemma 3·1 and Lemma 4·1, it is valid to interchange the order of summation and integration, and so this becomes
For each $j=1,\dots,k,$ we decompose the vertical contour $ I_j\;:\!=\;\left\{s_j \in \mathbb{C}\, : \,\mathrm{Re}(s_j)=1+{1}/{2\log x_j}\right\}$ as $I_{j}^{(1)} \cup I_{j}^{(2)} \cup I_{j}^{(3)}$ (traversed upwards), where
and
To establish an upper bound on the second derivative of the multiple Dirichlet series, we shall apply Cauchy’s integral formula for derivatives of k variables. For this purpose, we invoke Lemma 3·3 with the classical zero-free region
with $c=1/100$ say, for $j=1,\ldots,k.$ Moreover, we introduce the distinguished boundary
as there are various bounds on $\zeta(w_j)$ depending on the height. Then, Cauchy’s formula implies
with
given by (3·1) from Lemma 3·3. Using (3·3), [ Reference Titchmarsh21 , theorem 3·5] that
whenever ${1}/{4}\leqslant|\mathrm{Im}(w_j)| \leqslant T_j^2,$ and the simple upper bound
we arrive at the derivative bound
Applying (4·6) with $j=1,2$ from Lemma 4·1, for $j=1,\dots,k,$ we have the estimates
and
Therefore, combining with (4·5) from Lemma 4·1 and (4·9), the main contribution to (4·8) is
with an error term
as $T_j= 2(\!\log x_j)^{2}$ for $j=1,\ldots,k.$
Applying Lemma 3·4, the main contribution to (4·11) is
with an error term
For $j=1,\dots,k,$ we have
Combining with (4·10), the expression (4·14) is
Since for $j=1,\dots,k$ we have the bound
it follows from (4·10) that the main contribution to (4·13) is
with an error term
Applying Lemma 4·2, for $j=1,\dots,k$ we have
Finally, the lemma follows from collecting the main term (4·16) and the error terms (4·12), (4·15) and (4·17).
To proceed to the computation of the main term (4·1), we first show that it suffices to limit ourselves to the region where $u_1+\cdots+u_{k-1} \leqslant 1-{1}/{\log x}.$ Otherwise, if $u_1+\cdots+u_{k-1} > 1-{1}/{\log x},$ then we may assume without loss of generality that $u_{k-1}\geqslant {1}/{2k}.$ We now show that when replacing $u_{k-1}$ by $u_{k-1}-{1}/{\log x},$ both the right-hand sides of (4·1) and (1·1) are changed by a negligible amount. Arguing similarly as before, we have
If $u_{k-1}\leqslant {1}/{2},$ then using [ Reference Koukoulopoulos14 , theorem 14·2], this is bounded by
Otherwise, again it follows from the observation (4·4) that (4·18) is
On the other hand, by making the change of variables $t_j=(1-t_{k-1})s_j$ for $j=1,\ldots, k-2,$ we have
Therefore, we can always assume $u_1+\cdots+u_{k-1} \leqslant 1-{1}/{\log x}.$ Arguing similarly, we can further limit ourselves to the smaller region where $u_1,\dots,u_{k-1} \geqslant {1}/{\log x}$ as well.
In order to apply Lemma 4·3, we express the main term (4·1) as
For $j=1,\dots,k-1,$ again it follows from [ Reference Koukoulopoulos14 , theorem 14·2] that
and similarly
By partial summation (or more precisely multiple Riemann–Stieltjes integration) and Lemma 4·3, the main term of (4·20) is
Finally, it remains to compute the integrals $I_1$ and $I_2.$
Lemma 4·4. The first integral $I_1$ equals
Proof. Making the change of variables $x_j=x^{t_j}$ for $j=1,\ldots,k-1$ , the integral $I_1$ becomes
Integrating by parts with respect to $t_k$ gives
Therefore, the contribution of the first term of (4·21) to the integral $I_1$ is
Note that
Without loss of generality, it suffices to bound the term where $j=k-1.$ Similar to (4·19), we have
On the other hand, the contribution of the second term of (4·21) to the integral $I_1$ is
If $u_j > {1}/{2k}$ for some $j=1,\ldots, k-1,$ then this is
as $k \geqslant 2$ . Otherwise, the contribution is
We also have
so that the contribution of the last term of (4·21) to the integral $I_1$ is
Making the change of variables $s=1-t_1-\cdots-t_{k-1},$ this is bounded by
Similar to (4·19), the integral in the parentheses is $\ll (1-s)^{-\frac{1}{k}},$ and so (4·25) is
Collecting the main term of (4·22) and the error terms (4·23), (4·24) and (4·26), the lemma follows.
Lemma 4·5. The second integral $I_2$ is
Proof. The integral $I_2$ is bounded by
where
By integration by parts, for each $\boldsymbol{l},$ the integral $I_{2}^{(\boldsymbol{l})}$ is bounded by
For each subset $J \subseteq [k],$ the integral $I_{2}^{(\boldsymbol{l};J)}$ is
Applying Lemma 4·3, this is
Summing over every $\boldsymbol{l},$ we have
To avoid repetitions, we only bound the contribution of the term where $i=k$ here. Making the change of variables $x_j=x^{t_j}$ for $j=1,\ldots,k-1$ , it becomes
which is
Integrating by parts with respect to $t_k$ gives
Therefore, the expression (4·27) is
Similar to (4·19), this is
Finally, the lemma follows from summing over every subset $J \subseteq [k].$
5. Proof of Theorem 1·2
We first define the function field analogue of the multiple Dirichlet series $\mathcal{D}(s_1,\dots,s_k).$
Definition 5·1. For $(s_1,\dots,s_k) \in \Omega$ , the multiple Dirichlet series $\mathcal{D}_{\mathbb{F}_q[x]}(s_1,\dots,s_k)$ is defined as
Having the multiple Dirichlet series $\mathcal{D}_{\mathbb{F}_q[x]}(s_1,\dots,s_k)$ in hand, it is now clear that we can follow exactly the same steps as before. Moreover, since the function field zeta function $\zeta_{\mathbb{F}_q}(s)$ never vanishes (see [ Reference Rosen18 , chapter 2]), some of the computations above can be simplified considerably. To avoid repetitions, the complete proof is omitted here.
6. Proof of Theorem 1·3
It is clear that one can argue similarly but a more direct and elementary proof is presented here. We begin with the combinatorial analogue of the mean of divisor functions.
Lemma 6·1. Let $\alpha \in \mathbb{C}\setminus \mathbb{Z}_{\leqslant 0}$ and $n\in \mathbb{Z}_{\geqslant 0}.$ Then we have
Moreover, we have the estimate
Proof. Although (6·1) is fairly standard, say for instance one may apply [ Reference Stanley19 , corollary 5·1·9] with $f \equiv \alpha$ , we provide a short proof here for the sake of completeness. Adopting the notations in Section 2, we write
Now any permutation $\sigma \in S_n$ with k disjoint cycles can be constructed by the following procedure. To begin with, there are $(n-1)(n-2)\cdots(n-i_1+1)$ ways of choosing $i_1-1$ distinct integers from $[n]\setminus\{1\}$ to form a cycle $C_1$ with length $i_1$ containing $1.$ Then, fix any integer $m \in [n]$ not contained in the cycle $C_1$ . Similarly, there are $(n-i_1-1)\cdots(n-i_1-i_2+1)$ ways of choosing $i_2-1$ integers from $[n]\setminus C_1$ to form another cycle $C_2$ with length $i_2$ containing m. Repeating the same procedure until $i_1+\cdots+i_k=n$ , we arrive at a permutation with k disjoint cycles.
Therefore, the expression (6·1) follows from the explicit formula
which can be seen as the coefficient of ${\alpha}^k$ in the falling factorial
On the other hand, to prove (6·2), we express the binomial coefficient as a ratio of gamma functions, followed by the application of Stirling’s formula.
Similar to Theorem 1·1, without loss of generality we shall assume $u_1,\ldots u_{k-1}, 1-u_1-\cdots-u_{k-1} \geqslant {1}/{n}.$ Interchanging the order of summation, we have
where $m_k\;:\!=\;n-m_1-\cdots -m_{k-1}.$
Note that $\tau_k(\sigma)^{-1}=\tau_{1/k}(\sigma).$ Applying (6·1) from Lemma 6·1, the expression (6·3) equals
Let $I \subseteq [k-1]$ be a nonempty subset. Then using (6·2) from Lemma 6·1, the contribution of $m_i=0$ to (6·4) for $i\in I$ is
which is
Also, the contribution of $m_j>nu_j-1$ for some $j=1,\dots,k-1$ to (6·4) given that $m_1,\dots,m_k\geqslant 1$ is
Since $u_j \geqslant {1}/{n}$ for $j=1,\dots,k-1,$ this is
Collecting the error terms (6·5) and (6·6), the expression (6·4) equals
Applying (6·2) from Lemma 6·1, the main term of (6·7) is the Riemann sum
with an error term
Let us first bound the error term (6·9). For each $j=1,\dots,k-1,$ we have
Arguing similarly for $j=k,$ we also have
Therefore, the error term (6·9) is $\ll n^{-\frac{1}{k}}$ and we are left with the main term (6·8).
The distribution function $F(u_1,\ldots,u_{k-1})$ equals
The first error term in (6·10) is
The second error term in (6·10) is
By Taylor’s theorem, for $(t_1,\dots,t_{k-1}) \in\left[{m_1}/{n}, ({m_1+1})/{n} \right]\times \cdots \times [({m_{k-1}})/{n}, ({m_{k-1}+1})/{n}],$ we have
Using the approximation, we conclude from (6·10), (6·11) and (6·12) that
and the last error term here is exactly the same as (6·9), which is again $\ll n^{-\frac{1}{k}}.$
7. Factorisation into k parts in the general setting
With a view to model Dirichlet distribution with arbitrary parameters, we further explore the factorisation of integers into k parts in the general setting using multiplicative functions of several variables defined below.
Definition 7·1. An arithmetic function of k variables $F \colon \mathbb{N}^k \to \mathbb{C}$ is said to be multiplicative if it satisfies the condition $F(1,\ldots,1)=1$ and the functional equation
whenever $(m_1\cdots m_k, n_1\cdots n_k)=1,$ or equivalently,
where $v_p(n)\;:\!=\;\max\{k \geqslant 0 \, : \, p^k | n\}.$
Remark 7·1. Multiplicative functions of several variables, such as the “GCD function” and the “LCM function” are interesting for their own sake. See [ Reference Tóth22 ] for further discussion.
To adapt the proof of Theorem 1·1, we consider the following class of multiplicative functions.
Definition 7·2. Let $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_k), \boldsymbol{\beta}=(\beta_1,\ldots,\beta_k), \boldsymbol{c}= (c_1,\ldots,c_k),\boldsymbol{\delta}=(\delta_1,\ldots,\delta_k)$ with $\alpha_j, \beta_j, c_j>0, \delta_j\geqslant 0$ for $j=1,\ldots,k.$ We denote by $\mathcal{M}(\boldsymbol{\alpha};\boldsymbol{\beta},\boldsymbol{c}, \boldsymbol{\delta})$ the class of non-negative multiplicative functions of k variables $F \colon \mathbb{N}^k \to \mathbb{C}$ satisfying the following conditions:
-
(a) (divisor bound) for $j=1,\ldots,k,$ we have $|F(1,\ldots,\overbrace{n}^{\boldsymbol{j}-th},\ldots 1)| \leqslant \tau_{\beta_j}(n),$ where
\begin{align*}\tau_{\beta}(n)\;:\!=\;\prod_{p}\left(\substack{v_p(n)+\beta-1 \\[2pt] v_p(n)}\right)\end{align*}is the generalised divisor function; -
(b) (analytic continuation) let $s=\sigma+it \in \mathbb{C}.$ For $j=1,\ldots,k$ , the Dirichlet series
-
(c) (growth rate) for $j=1,\ldots,k,$ in the domain above we have the bound
\begin{align*}\mathcal{P}_F(s;\boldsymbol{\alpha},j) \leqslant \delta_j\log (2+|t|).\end{align*}
For instance, the multiplicative function $F(n_1,\ldots,n_k)=\tau_k(n_1\cdots n_k)^{-1}$ belongs to the class
Applying the Mellin transform to (higher derivatives of) the multiple Dirichlet series
as before, one can prove the following generalisation of Lemma 4·3.
Lemma 7·1. Given a multiplicative function of k variables $ F \in \mathcal{M}(\boldsymbol{\alpha};\boldsymbol{\beta}, \boldsymbol{c}, \boldsymbol{\delta}).$ Let $m \geqslant 2$ be an integer and $x_1,\dots,x_k \geqslant e.$ We denote by $S_F(x_1,\dots,x_k;m)$ the weighted sum
Then there exists $m_0=m_0(\boldsymbol{\alpha},\boldsymbol{\beta}, \boldsymbol{c}, \boldsymbol{\delta})$ such that for any integer $m\geqslant m_0,$ we have
with
To model the Dirichlet distribution by factorizing integers into k parts, we consider the following class of pairs of multiplicative functions.
Definition 7·3. Let $\theta>0$ and $\boldsymbol{\alpha}$ be a positive k-tuple. We denote by $\mathcal{M}_{\theta}(\boldsymbol{\alpha})$ the class of pairs of multiplicative functions $(f; G) $ satisfying the following conditions:
-
(a) for $n \geqslant 1,$ we have
\begin{align*}\sum_{n=d_1\cdots d_k}G(d_1,\ldots,d_k)>0;\end{align*} -
(b) the multiplicative function f belongs to the class $\mathcal{M}(\theta;\beta',c', \delta' )$ for some $\beta', c', \delta';$
-
(c) the multiplicative function of k variables
\begin{align*}F(d_1,\ldots,d_k)\;:\!=\;f(n)\cdot\frac{G(d_1,\ldots,d_k)}{\sum_{n=e_1\cdots e_k}G(e_1,\ldots,e_k)}\end{align*}belongs to the class $\mathcal{M}(\boldsymbol{\alpha};\boldsymbol{\beta}, \boldsymbol{c}, \boldsymbol{\delta})$ for some $\boldsymbol{\beta}, \boldsymbol{c}, \boldsymbol{\delta},$ where $n=d_1\cdots d_k.$
Remark 7·2. By definition, we must have $\theta=\alpha_1+\cdots+\alpha_k.$
Then, applying Lemma 7·1 followed by partial summation as before, one can prove the following generalisation of Theorem 1·1.
Theorem 7·1. Let $(f;G)$ be a pair of multiplicative functions belonging to the class $\mathcal{M}_{\theta}(\boldsymbol{\alpha}).$ Then uniformly for $x \geqslant 2$ and $u_1,\ldots,u_{k-1} \geqslant 0$ satisfying $u_1+\cdots+u_{k-1}\leqslant 1,$ we have
Finally, we conclude with the following generalisation of Corollary 1·1.
Corollary 7·1. Given a pair of multiplicative functions $(f;G)$ belonging to the class $\mathcal{M}_{\theta}(\boldsymbol{\alpha}).$ For $x\geqslant 1,$ let n be a random integer chosen from [1, x] with probability $\left(\sum_{m \leqslant x}f(m)\right)^{-1}f(n)$ and $(d_1,\ldots,d_k)$ be a random k-tuple chosen from the set of all possible factorisation $\{(m_1,\ldots,m_k) \in \mathbb{N}^k\, : \, n=m_1\cdots m_k\}$ with probability $\left( \sum_{n=e_1\cdots e_k}G(e_1,\ldots,e_k) \right)^{-1}G(d_1,\ldots,d_k)$ . Then as $x \to \infty,$ we have the convergence in distribution
Remark 7·3. See [ Reference Bareikis and Mac̆iulis2, Reference Bareikis and Mac̆iulis3 ] for the cases where $k=2,3$ respectively, where $G(d_1,\ldots,d_k)$ is of the form $(f_1\ast \cdots \ast f_{k-1} \ast 1)(d_1\cdots d_k)$ for some multiplicative functions $f_1,\ldots,f_{k-1}\;:\;\mathbb{N} \to \mathbb{C}.$
Example 7·1. For $k\geqslant 2,$ let $\theta, \lambda_1, \ldots, \lambda_k>0.$ We consider the pair of multiplicative functions
Then the Dirichlet distribution of dimension k
can be modelled in the sense of Corollary 7·1. In particular, when $\theta, \lambda_1,\ldots, \lambda_k=1,$ it reduces to Theorem 1·1.
Example 7·2. For $q\geqslant 3,$ let $\{a_1,\ldots,a_{\varphi(q)}\}$ be a reduced residue system $\ (\mathrm{mod}\ q)$ . We consider the pair of multiplicative functions
Then the Dirichlet distribution of dimension $\varphi(q)$
can be modelled in the sense of Corollary 7·1. In particular, when $q=4,$ it reduces to [ Reference Montgomery and Vaughan15 , exercise 6·2·22].
Example 7·3. For $k\geqslant 2,$ we consider the pair of multiplicative functions
Then the Dirichlet distribution of dimension k
can be modelled in the sense of Corollary 7·1. In particular, when $k=2,$ it reduces to [ Reference Daoud, Hidri and Naimi6 , theorem 2].
Example 7·4. For $k\geqslant 2,$ we consider the pair of multiplicative functions
Then the Dirichlet distribution of dimension k
can be modelled in the sense of Corollary 7·1. In particular, when $k=2,$ it reduces to [ Reference Feng and Cui8 , theorem 2] with $y=x.$
Example 7·5. For $k\geqslant 2$ , let $\mathcal{R}$ be a subset of $\{\{i,j\}\,:\, 1 \leqslant i \neq j \leqslant k\}.$ We consider the pair of multiplicative functions
Then the Dirichlet distribution of dimension k
can be modelled in the sense of Corollary 7·1. In particular, when $k=2^r$ for $r \geqslant 2,$ it reduces to [ Reference de la Bretèche and Tenenbaum4 , théorème 1·1] with a suitable subset $\mathcal{R}$ via total decomposition sets (see [ Reference Hall12 , theorem 0·20]), which is itself a generalisation of [ Reference Bareikis and Mac̆iulis1 , theorem 2·1] for $r=2.$
Example 7·6. For $k\geqslant 3$ , we consider the pair of multiplicative functions
Then the Dirichlet distribution of dimension k
can be modelled in the sense of Corollary 7·1. In particular, when $k=3,$ it reduces to [ Reference de la Bretèche and Tenenbaum4 , théorème 1·2].
Unsurprisingly, we expect that Theorem 7·1 should also hold for polynomials or permutations. Specifically, in the realm of permutations, the counterpart to multiplicative functions is the generalised Ewens measure (see [ Reference Elboim and Gorodetsky7 ]). Detailed proofs will be provided in the author’s doctoral thesis.
Acknowledgements
The author is grateful to Andrew Granville and Dimitris Koukoulopoulos for their suggestions and encouragement. He would also like to thank Sary Drappeau for pointing out relevant papers, and the anonymous referee for helpful comments and corrections.