The least common multiple of sets of positive integers


Ana Zumalacarregui




Wed, 18/03/2015 - 1:30pm


OMB 145 (Old Main Building)


It is well known that the classical Chebyshev's function $\psi(n)$ has an alternative expression in terms of the least common multiple of the first n integers: $\psi(n) = \log \text{lcm} (1,2, ... , n)$.

Here we generalize this function by considering, for a set A of $\{1,..., n\}$, the quantity  $\psi(A) := \log  \text{lcm} \{a : a \in A\}$ and we ask ourselves about its asymptotic behavior.

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

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

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