Limit Theorems for Empirical Density of Greatest Common Divisors

Mehrdad, Behzad; Zhu, Lingjiong

doi:10.1017/s0305004116000426

Cited by 6 publications

(7 citation statements)

References 13 publications

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“…[29] and [12]. Further refinements of these celebrated results can be found in the recent papers [18,25,26].…”

Section: Introductionmentioning

confidence: 72%

“…Moreover, the events (C j,p,n ) j=1,...,k are disjoint and equiprobable. Thus, the limit Keeping in mind that λ p (X (n) ) = 0 for p > n, we see that that it is enough to prove that (25) lim…”

Section: Proofsmentioning

confidence: 88%

“…[29] and [12]. Further refinements of these celebrated results can be found in the recent papers [18,25,26]. Cesàro also considered similar questions when the greatest common divisor (gcd) is replaced by the least common multiple (lcm).…”

Section: Introductionmentioning

confidence: 94%

“…The first probability can be estimated as follows Keeping in mind that λ p (X (n) ) = 0 for p > n, we see that that it is enough to prove that (25) It remains to apply formula (6) in [16], which is an extension of (27), to obtain…”

Section: Proofsmentioning

confidence: 99%

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On the least common multiple of several random integers

Bostan

Marynych

Raschel

2019

Journal of Number Theory

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Let Ln(k) denote the least common multiple of k independent random integers uniformly chosen in {1, 2, . . . , n}. In this note, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as n → ∞ of f (Ln(k)) n rk for a wide class of multiplicative arithmetic functions f with polynomial growth r > −1. Furthermore, we identify the limit as an infinite product of independent random variables indexed by prime numbers. Along the way, we compute the generating function of a trimmed sum of independent geometric laws, occurring in the above infinite product. This generating function is rational; we relate it to the generating function of a certain max-type Diophantine equation, of which we solve a generalized version. Our results extend theorems by Wintner (1939), Fernández andFernández (2013) and Hilberdink and Tóth (2016).

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“…[29] and [12]. Further refinements of these celebrated results can be found in the recent papers [18,25,26].…”

Section: Introductionmentioning

confidence: 72%

Section: Proofsmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 94%

Section: Proofsmentioning

confidence: 99%

See 2 more Smart Citations

On the least common multiple of several random integers

Bostan

Marynych

Raschel

2019

Journal of Number Theory

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“…Now large deviations for independent and non-identically distributed random variables have been readily available since the 1970's, thus this author certainly finds it quite surprising that until very recently no one extended the parallel between (1.2) and (1.4), or (1.1) and (1.3), to study the large deviations of (1.2) or (1.1)! See [16,17].…”

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confidence: 99%

A Natural Probabilistic Model on the Integers and Its Relation to Dickman-Type Distributions and Buchstab’s Function

Pinsky

2019

Springer Proceedings in Mathematics &Amp; Statistics

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Let {pj } ∞ j=1 denote the set of prime numbers in increasing order, let ΩN ⊂ N denote the set of positive integers with no prime factor larger than pN and let PN denote the probability measure on ΩN which gives to each n ∈ ΩN a probability proportional to 1 n . This measure is in fact the distribution of the random integer IN ∈ ΩN definedare independent random variables and Xp j is distributed as Geom(1 − 1 p j ). We show that log n log N under PN converges weakly to the Dickman distribution. As a corollary, we recover a classical result from multiplicative number theory-Mertens'We show that the two densities coincide on a natural algebra of subsets of N. We also show that they do not agree on the sets of n 1 s -smooth numbers {n ∈ N : p + (n) ≤ n 1 s }, s > 1, where p + (n) denotes the largest prime divisor of n. This last consideration concerns distributions involving the Dickman function. We also consider the sets of n 1 s -rough numbers {n ∈ N : p − (n) ≥ n 1 s }, s > 1, where p − (n) denotes the smallest prime divisor of n. We show that the probabilities of these sets, under the uniform distribution on [N ] = {1, . . . , N } and under the PN -distribution on ΩN , have the same asymptotic decay profile as functions of s, although their rates are necessarily different. This profile involves the Buchstab function. We also prove a new representation for the Buchstab function.

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Wasserstein distance, Fourier series and applications

Steinerberger

2021

Monatsh Math

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Limit Theorems for Empirical Density of Greatest Common Divisors

Cited by 6 publications

References 13 publications

On the least common multiple of several random integers

On the least common multiple of several random integers

A Natural Probabilistic Model on the Integers and Its Relation to Dickman-Type Distributions and Buchstab’s Function

Wasserstein distance, Fourier series and applications

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