The least common multiple of sets of positive integers

Abstract: 

It is well known that the classical Chebyshev's function ψ(n)ψ(n) has an alternative expression in terms of the least common multiple of the first n integers: ψ(n)=loglcm(1,2,...,n)ψ(n)=log⁡lcm(1,2,...,n).

Here we generalize this function by considering, for a set A of {1,...,n}{1,...,n}, the quantity  ψ(A):=loglcm{a:a∈A}ψ(A):=log⁡lcm{a:a∈A} and we ask ourselves about its asymptotic behavior.

We will focus on sets given by Af={f(1),f(2),...,f(n)}Af={f(1),f(2),...,f(n)} for some polynomial with integer coefficients and also discuss the case where the set is chosen at random in {1,...,n}{1,...,n}, by considering two different models, analogous to G(n,p)G(n,p) and G(n,M)G(n,M) models for random graphs.

Joint work with J. Cilleruelo, J. Rue and P. Sarka.