The least common multiple of random sets of positive integers

Cilleruelo, Javier; Rué, Juanjo; Šarka, Paulius; Zumalacárregui, Ana

doi:10.1016/j.jnt.2014.04.011

Cited by 23 publications

(22 citation statements)

References 3 publications

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“…Properties of the operation m • n = [m, n]/(m, n) were investigated by the first author [14]. Note that the following recent result of different type, concerning the lcm of several positive integers, was obtained by J. Cilleruelo, J. Rué, P.Sarka and A. Zumalacárregui [8]: lcm{a : a ∈ A} = 2 n(1+o(1)) for almost all subsets A ⊂ {1, . .…”

Section: Introductionmentioning

confidence: 99%

On the average value of the least common multiple of k positive integers

Hilberdink

Tóth

2016

Journal of Number Theory

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Section: Introductionmentioning

confidence: 99%

On the average value of the least common multiple of k positive integers

Hilberdink

Tóth

2016

Journal of Number Theory

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“…, n} with probability δ. Motivated by (1), Cilleruelo, Rué,Šarka, and Zumalacárregui [8] proved the following result (see also [5] for a more precise version, and [6,7] for others results of similar flavor). Theorem 1.1.…”

Section: Introductionmentioning

confidence: 93%

On the l.c.m. of random terms of binary recurrence sequences

Sanna

2020

Journal of Number Theory

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For every positive integer n and every δ ∈ [0, 1], let B(n, δ) denote the probabilistic model in which a random set A ⊆ {1, . . . , n} is constructed by choosing independently every element of {1, . . . , n} with probability δ. Moreover, let (u k ) k≥0 be an integer sequence satisfying u k = a1u k−1 + a2u k−2 , for every integer k ≥ 2, where u0 = 0, u1 = 0, and a1, a2 are fixed nonzero integers; and let α and β, with |α| ≥ |β|, be the two roots of the polynomial X 2 − a1X − a2. Also, assume that α/β is not a root of unity.We prove that, as δn/ log n → +∞, for every A in B(n, δ) we have log lcm(ua : a ∈ A) ∼ δ Li2(1 − δ)

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“…We will also use the two Chebyshev functions ϑ and ψ. The first Chebyshev function ϑ : R → R is defined as Recalling the identity log LCM ([n]) = ψ(n), we state the following result taken from [4], see Lemma 2.1 therein.…”

Section: Preliminariesmentioning

confidence: 99%

Limit theorems for the least common multiple of a random set of integers

Alsmeyer

Kabluchko

Marynych

2019

Trans. Amer. Math. Soc.

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Let Ln be the least common multiple of a random set of integers obtained from {1, . . . , n} by retaining each element with probability θ ∈ (0, 1) independently of the others. We prove that the process (log L nt ) t∈[0,1] , after centering and normalization, converges weakly to a certain Gaussian process that is not Brownian motion. Further results include a strong law of large numbers for log Ln as well as Poisson limit theorems in regimes when θ depends on n in an appropriate way.2010 Mathematics Subject Classification. Primary: 60F05; secondary 11N37, 60F15. Keywords: random set of integers, least common multiple, law of large numbers, central limit theorem, functional limit theorem, Gaussian process, von Mangoldt function, Chebyshev functions.

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The least common multiple of random sets of positive integers

Abstract: Abstract. We study the typical behavior of the least common multiple of the elements of a random subset A ⊂ {1, . . . , n}. For example we prove that lcm{a : a ∈ A} = 2 n(1+o(1)) for almost all subsets A ⊂ {1, . . . , n}.

Cited by 23 publications

References 3 publications

On the average value of the least common multiple of k positive integers

On the average value of the least common multiple of k positive integers

On the l.c.m. of random terms of binary recurrence sequences

Limit theorems for the least common multiple of a random set of integers

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