Divisibility properties of random samples of integers
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Divisibility properties of random samples of integers José L. Fernández1 · Pablo Fernández1 Received: 12 November 2019 / Accepted: 28 October 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract This paper is devoted to survey the rich theory, some of it quite recent, concerning the divisibility properties, of various kinds, of random r -tuples of positive integers. Keywords Divisibility · Random samples of integers · Distribution and moments of gcd and lcm · Coprimality and pairwise coprimality · Asymptotic normality · Visible points · Random walk · Waiting times Mathematics Subject Classification 11K65 · 11N37 · 11A25 · 60E05
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Some notations . . . . . . . . . . . . . . . . . . . . . . . 1.2 Primes play a game of chance . . . . . . . . . . . . . . . . 1.3 Some other probabilities . . . . . . . . . . . . . . . . . . 2 Statistics of the greatest common divisor . . . . . . . . . . . . 3 Statistics of the least common multiple . . . . . . . . . . . . . 3.1 Moments of the least common multiple . . . . . . . . . . . 3.2 Distribution of the least common multiple . . . . . . . . . 4 Coprimality and codivisibility properties of samples of integers 4.1 Pairwise and k-wise coprime tuples of integers . . . . . . . 4.2 Algebraic number fields: ideal and algebraic integers . . . 4.3 Index of codivisibility . . . . . . . . . . . . . . . . . . . . 4.4 Number of coprime pairs and other statistics . . . . . . . . 5 The distribution of visible points in the 2-dimensional lattice . . 5.1 Hidden squares . . . . . . . . . . . . . . . . . . . . . . . 5.2 Realizable patterns . . . . . . . . . . . . . . . . . . . . . 5.3 Random walks in the lattice . . . . . . . . . . . . . . . . . 5.4 Windows and visibility . . . . . . . . . . . . . . . . . . . 6 Waiting times . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The research of both authors is partially supported by Fundación Akusmatika.
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Pablo Fernández [email protected] José L. Fernández [email protected]
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Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain 0123456789().: V,-vol
123
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J. L. Fernández, P. Fernández
6.1 Waiting
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