Modeling the Dirichlet distribution using multiplicative functions

. For q, m, n, d ∈ N and some multiplicative function f (cid:62) 0 , we denote by T 3 ( n ) the sum of f ( d ) over the ordered triples ( q, m, d ) with qmd = n . We prove that Cesaro mean of distribution functions deﬁned by means of T 3 uniformly converges to the one-parameter Dirichlet distribution function. The parameter of the limit distribution depends on the values of f on primes. The remainder term is estimated as well.

It is known various high efficiency algorithms for generating the Dirichlet random vectors (see, e.g., [6,10,11]). Usually, in this case, the efficiency is empirically estimated in terms of computer generation time.
In the probabilistic number theory, we have a slightly different problem. We need to construct a sequence of vectors, whose "average" distribution function tends to Dirichlet distribution. These "arithmetical" vectors are supposed to be related to the arithmetical functions. In this paper, we consider the construction of two-dimensional "arithmetical" vectors and the convergence of their distributions to some given Dirichlet distribution.
In the one-dimensional case, the first attempt to simulate B(1/2, 1/2), that is, the Arcsine law, by means of the divisor function was made in [9]. Manstavičius [12] noticed that not only the Arcsine law but also some other distributions can occur as a limits. Some ideas of these papers were extensively used for the simulation of some Beta related distributions by means of the divisor function with multiplicative weight (see [1,2,4,5,7,8]). General result of this type for any Beta distribution B(a, b) was obtained in [3].
The number of ordered factorisations of n ∈ N into three factors is known as function τ 3 (n) := l1l2l3=n 1. For any multiplicative function f : N → R, we define where τ (n) := d|n 1 is the classical divisor function. If f (n) ≡ 1, then T 3 (n) = τ 3 (n). Let (X n ; Y n ) be the random vector, which takes values (ln d 1 / ln n; ln d 2 / ln n) when (d 1 ; d 2 ) run trough all ordered pairs of divisors of n with uniform probability 1/T 3 (n).

Its distribution function is
This sequence of distributions does not converge pointwise on [0, 1] × [0, 1] (see [13]). Therefore, following [9], the corresponding Cesaro mean can be considered. When f ≡ 1, Nyandwi and Smati [13] proved that S x (u, v) tends to the Dirichlet distribution function D(u, v; 1/3, 1/3, 1/3). In this paper, we generalize the result of [13] showing that some class of the one-parameter Dirichlet distributions can be simulated by S x (u, v). The parameter of the limit distribution depends on the values of multiplicative function f on primes. More precisely, we assume that the values f (p) satisfy some regularity conditions so that the multiplicative function 1/T 3 belongs to the class G: Definition 1. We say that multiplicative function ϕ : N → [0; ∞) belongs to the class G(κ, δ) for some constants κ, δ 0 if ϕ(p k ) C for all primes p and k ∈ N and the function L(s) : for some 0 < c 0 1/2, has an analytic continuation P (s) into the region where P (s) is holomorphic, and |P (s)| δ log(|τ | + 1) + c 1 with some c 0 0.
We assume that c and C with or without subscripts denote constants. The aim of this paper is to prove the following: Theorem 1. Let f be a nonnegative multiplicative function such that 1/T 3 ∈ G(α, δ), 0 < α < 1/2 and 0 δ < 1/2. Then for all u, v ∈ [0, 1], Here Unless otherwise indicated, here and in what follows, we assume that x → ∞, the implicit constants in the or O(·) symbols depend at most on the parameters and constants involved in the definitions of the corresponding classes G(·).

Preliminaries
For κ > 0 and any multiplicative function θ, set Here and in what follows, we assume that p is prime. A slight modification to the proof of Lemma 3.1 in [2] yields Lemma 1. Let ϕ and g be the nonnegative multiplicative functions such that for some C 1 > 0 and all integers j, k with 0 j k. Assume furthermore that ϕ · g ∈ G(κ, δ), κ > 0 and 0 δ < 1. Then, uniformly for all x 1 and d ∈ N, Here σ 0 = σ(0), and c 2 0 is a constant depending on c 0 , κ and C 1 .
This sum may be evaluated in terms of the integral provided some information about the behaviour of the sum The following consequence of Lemmas 3 and 4 in [3] will be applied to evaluate the sum Θ x (u, w, b).
The implicit constant in symbol depends at most on a and b.
We will need some estimates of the integrals The implied constant in depending on a, b, c only.
Proof. Let us begin with the proof of (4). By the definition, for 0 u < 1, This implies (4) since b + c < 2. Further, observe that and Therefore (3) Here the implied constant in depending at most on a, b, c.
For ε < u 1 − 2ε, we have where From this, (13), (16) and (17) it follows For ∆ 2 (ε, u), we employ the Lagrange mean value theorem again and obtain the expression similar to that in (15) with integrals J 2 instead of J 1 . Then we estimate integrals J 2 by means of Lemma 6 and find that When 1 − 2ε < u < 1, we have 2ε, a, b, c) .
(i) Let min(u, v) η x . Then we apply (23) and (25) to obtain To evaluate Z x (q, q, v, b; θ), we will employ Lemma 8. Note that for q x u and 0 s v, Therefore by Lemma 8 In view of (25), the remainder term in (27) is ln −a x + ln 1−a−b x. We deal with the main term using Lemma 8 again: Therefore, changing the integration variable t by z = t(1 − s), we transform (28) as follows: http://www.journals.vu.lt/nonlinear-analysis It remains to estimate the integral Analogously, I(a, a + 1) ln a x. Hence the first assertion of lemma follows from (29), (27) together with (26).
From what we have already established Therefore (24) follows from Lemma 5.

Partial summation gives
Lemma 2 and (10) yield Evaluating the derivative in H 3 , we obtain In view of Lemma 2, (0, η x , u, a, a, b + 1) ln a x .
The first integral of the last relation is 1, and the second one we estimate using Lemma 6. This gives It remains to evaluate the second term in (33). Using Lemma 1, we get A(1 − a, θ 1 ) ln a−1 x J 2 (0, η x , u, a, a, b) + O J 2 (0, η x , u, a + 1, a, b) ln a x .
Combining the last two relations with (45) and having in mind (44) and (42), we obtain The proof of Theorem 1 follows now from this estimate, (43), (36) and (34).