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Section: New Results

Least common multiple of random integers

In [4], Alin Bostan together with Kilian Raschel (CNRS, Tours) and Alexander Marynych (U. Kyiv, Ukraine) have investigated the least common multiple of random integers. Using a purely probabilistic approach, they derived a criterion for the convergence in distribution as n of f(Ln)/nrk, for a wide class of multiplicative arithmetic functions f with polynomial growth r, where Ln(k) denotes the least common multiple of k independent random integers with uniform distribution on {1,2,...,n}. Furthermore, they identified the limit as an infinite product of independent random variables indexed by the prime numbers. Along the way of showing the main results, they computed the (rational) generating function of a trimmed sum of independent geometric laws, which appears in the above infinite product. The latter is directly related to the generating function of a certain max-type diophantine equation, of which they solved a generalized version. The results extend theorems by Erdős and Wintner (1939), Fernández and Fernández (2013) and Hilberdink and Tóth (2016).