Note on the prime divisors of Farey fractions

. Let P 1 ( n ) > P 2 ( n ) > · · · be the prime divisors of a natural number n arranged in the non-increasing order. The limit distribution of the sequences ( log P i ( mn ) / log( mn ) , i > 1 ) for m/n ∈ ( λ 1 ; λ 2 ) , n 6 x , are considered. It is proved that under some conditions on λ i the limit distribution of the sequences exists and is closely related to the Poisson–Dirichlet distribution.


Introduction and the main result
Let N denote the set of natural numbers and R ∞ the linear space of all real sequences x = (x 1 , x 2 , . . . ) endowed with the product topology. It is well known that R ∞ is a separable metrizable topological space. Consider the function ξ : N → R ∞ defined as follows: if n = p 1 · · · p t with all p i primes and p 1 · · · p t then ξ(n) = (log p 1 , . . . , log p t , 0, 0, . . . ).
Let ν x denote the uniform distribution on {n ∈ N: n x}. The function n → n is a random variable on the probability space (N, ν x ); we denote it by the same letter n. Then ξ(n) is a random element of R ∞ defined on (N, ν x ).
It was proved by P. Billingsley in [1] that here denotes convergence in distribution (as x → ∞) and η is a random element of R ∞ distributed accordingly to the so-called Poisson-Dirichlet law. The new proof of this fact was given by P. Donnelly and G. Grimmett in [4].
We set the analogous problem of convergence of probabilistic measures, related to rational numbers.
Let Q + denote the set of positive rational numbers, I ⊂ (0; ∞) and ν I x denote the uniform distribution on Each element of Q + is represented in the unique way by an irreducible fraction m/n; we consider the nominator and denominator of it as random variables on the probability space (Q + , ν I x ), denoted by the same letters m and n. The following theorem was proved by the second author in [7] using the proof in [4] as a model. Theorem 1. Let I = (λ 1 ; λ 2 ), where 0 λ 1 < λ 2 < ∞ satisfy the condition: for an arbitrary 0 < γ 1 where η ′ is an independent copy of η.
In this paper we consider the limit distribution of ξ(mn) log(mn) . Let and R : ∆ → ∆ be the ranking function of Billingsley (see [2, Chapter 1, Section 4]). It omits the zero components of the infinite tuple and rearranges the positive ones into non-increasing order; if the resulting tuple is finite, the infinite tail of zeros is Our main result is the following theorem.
where η, η ′ are the same random elements as in Theorem 1.
Proof. Let m = p 1 · · · p s , n = q 1 · · · q t with all p i , q j primes, p 1 · · · p s and q 1 · · · q t . Then Since both R and T are continuous, the theorem follows from Theorem 1 and Lemma 1 below, which is proved in Section 3. ⊓ ⊔ Lemma 1. If conditions (1) are satisfied, then log n log(mn) It can be shown actually, that only the values p 1 can appear in (1).

Marginal distributions
Let P 1 (n) P 2 (n) · · · be the prime divisors of n arranged in the non-increasing order. Then the distributions of log P k (n)/ log n converge as x → ∞ to the onedimensional marginal distributions of the Poisson-Dirichlet law. Since log n/ log x 1, the same is true for the distributions of log P k (n)/ log x. The marginal distributions of the Poisson-Dirichlet measure in the number-theoretic context were discovered indeed in the form The investigation of these asymptotics was initiated by K. Dickman [3]. The properties of the function ρ(u) = ρ 1 (u) were investigated by N.G. de Bruijn. It is called Dickman-de Bruijn function and is defined by the following differential-delay equation: The papers of Ramaswami [6], Knuth and Trabb Pardo [5] followed, the functions ρ k (u) were investigated in numerous articles. It was shown, for example, that they are uniquely determined by the following properties: ρ k (u) = 1 for 0 u 1 and The multidimensional-marginal distributions are described by P. Billingsley [1], [2], see also A. Vershik [8]. They showed that ν x log P 1 (n) log n u 1 , . . . , log P k (n) log n u k → Φ k (u 1 , . . . , u k ), where the functions Φ k are expressed via the Dickman-de Bruijn function in the following way: In this section we find limit distributions for log P k (mn)/ log(mn), where m and n are random variables on (Q + , ν I x ). Suppose that conditions (1) are satisfied and denote α = p/(p + 1), β = 1 − α. Let η = (η 1 , η 2 , . . . ) and η ′ = (η ′ 1 , η ′ 2 , . . . ) be independent random sequences, distributed accordingly the Poisson-Dirichlet law, and ζ = (ζ 1 , ζ 2 , . . . ) = RT (αη, βη ′ ). Then, by Theorem 2, Let F k and G k denote the distribution functions of η k and ζ k , respectively. Then F k (u) = ρ k (1/u). We show how G k is expressed via F i with i k.
The case k = 1 is the most simple. Since ζ 1 = max(αη 1 , βη ′ 1 ), we have In the general case it is more convenient to work with G * k (u) = 1−G k (u) and F * k (u) = 1 − F k (u). For positive integers i, j define the random events U i0 = {αη i > u}, U 0j = βη ′ j > u , and U ij = αη i > u, βη ′ j > u . The event {ζ k > u} occurs if at least one of the events U ij with i + j = k appears. Hence The probabilities of the events U ij as well as of their intersections can be expressed via the functions F * k (u). Let us consider the case k = 2 for example. We have