On the density of coprime m-tuples over holomorphy rings

For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm x or less as x goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with explicit error terms. We also present results for higher moments along with computational data which support the results for the second, third, fourth, and fifth moments. The analogous question for integers is studied by Diaconis and Erdös.

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“…Of further interest are function field analogues. Some interesting results in this direction may be found in [9].…”

Section: Introductionmentioning

confidence: 91%

On the distribution of the greatest common divisor of Gaussian integers

Bradley

Cheng

Luo³

2016

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“…Of further interest are function field analogues. Some interesting results in this direction may be found in [Micheli and Schnyder 2015].…”

Section: Introductionmentioning

confidence: 91%

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2016

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Whole genome sequencing has revealed several examples where genomes of different species are related by permutation. The number of certain types of rearrangements needed to transform one permuted list into another can measure the distance between such lists. Using an algorithm based on three basic DNA editing operations suggested by a model for ciliate micronuclear decryption, this study defines the distance between two permutations to be the number of ciliate operations the algorithm performs during such a transformation. Combining wellknown clustering methods with this distance function enables one to construct corresponding phylogenies. These ideas are illustrated by exploring the phylogenetic relationships among the chromosomes of eight fruit fly (Drosophila) species, using the well-known UPGMA algorithm on the distance function provided by the ciliate operations.Over evolutionary time, "local" DNA editing events such as nucleotide substitutions, deletions or insertions diversify the set of DNA sequences present in organisms. Results of whole genome sequencing suggest that also "global" DNA editing events diversify these DNA sequences.Consider two species S 1 and S 2 with a common ancestor whose genome was organized over n linear chromosomes. A gene G of the ancestor was inherited as gene G 1 by species S 1 and as gene G 2 by species S 2 . G 1 and G 2 are orthologous genes, or simply orthologs. Assume that the species S 1 and S 2 each also has n chromosomes, and that for each ancestral chromosome i, the orthologs of any ancestral gene on chromosome i are also in the descendant species S 1 and S 2 on the corresponding chromosome i. This assumption is known, in the context of certain MSC2010: 05E15,

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“…[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%

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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On the density of coprime m-tuples over holomorphy rings

Cited by 10 publications

References 8 publications

On the distribution of the greatest common divisor of Gaussian integers

On the distribution of the greatest common divisor of Gaussian integers

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On the least common multiple of several random integers

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