The least common multiple of sets of positive integers

Details The least common multiple of sets of positive integers The least common multiple of sets of positive integers

2011-12-13

arxiv.org/abs/1112.3013v1

Mathematics

Number Theory

13 pages

Scientific paper

We prove that $\log lcm\{a\in A\}=n\log 2+o(n)$ for almost every set $A\subset \{1,..., n\}$. We also study the typical behavior of the logarithm of the least common multiple of sets of integers in $\{1,..., n\}$ with prescribed size. For example, we prove that, for any $0<\theta<1$, $\log lcm\{a\in A\}=(1-\theta)n^{\theta}\log n +o(n^{\theta})$ for almost all sets $A\subset\{1,...,n\}$ of size $\lfloor n^{\theta}\rfloor$. Extremal values of $\log \text{lcm}\{a\in A\}$ for sets $A$ of prescribed size are also studied.

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