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On the least common multiple of several random integers
- Source :
- Journal of Number Theory, Journal of Number Theory, 2019, 204, pp.113--133. ⟨10.1016/j.jnt.2019.03.017⟩, Journal of Number Theory, Elsevier, 2019, 204, pp.113--133. ⟨10.1016/j.jnt.2019.03.017⟩
- Publication Year :
- 2019
- Publisher :
- Elsevier BV, 2019.
-
Abstract
- Let $L_n(k)$ denote the least common multiple of $k$ independent random integers uniformly chosen in $\{1,2,\ldots ,n\}$. In this note, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as $n\to\infty$ of $\frac{f(L_n(k))}{n^{rk}}$ 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 Erd\H{o}s and Wintner (1939), Fern\'{a}ndez and Fern\'{a}ndez (2013) and Hilberdink and T\'{o}th (2016).<br />Comment: 19 pages
- Subjects :
- trimmed sums of geometric laws
Primary: 11A05, 11N37, Secondary: 11A25, 60F05
Convergence in distribution
least common multiple
Infinite product
010103 numerical & computational mathematics
prime products
01 natural sciences
Combinatorics
FOS: Mathematics
Arithmetic function
Number Theory (math.NT)
0101 mathematics
Least common multiple
Mathematics
Algebra and Number Theory
Mathematics - Number Theory
Diophantine equation
Probability (math.PR)
010102 general mathematics
Multiplicative function
Generating function
Prime number
[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Convergence of random variables
Mathematics - Probability
Subjects
Details
- ISSN :
- 0022314X and 10961658
- Volume :
- 204
- Database :
- OpenAIRE
- Journal :
- Journal of Number Theory
- Accession number :
- edsair.doi.dedup.....84b442f8f2823ff325b167973d59307c