Dense sets of natural numbers with unusually large least common multiples

Abstract

We construct a set $A \subset \mathbf{N}$ such that one has $$ \sum_{n \in A: n \leq x} \frac{1}{n} = \exp\left(\left(\frac{1}{2}+o(1)\right) (\log\log x)^{1/2} \log\log\log x \right)$$ and $$ \frac{1}{(\sum_{n \in A: n \leq x} \frac{1}{n})^2} \sum_{n,m \in A: n < m \leq x} \frac{1}{\operatorname{lcm}(n,m)} \asymp 1$$ for sufficiently large $x$. The exponent $\frac{1}{2}$ can replaced by any other positive constant, but the growth rate is otherwise optimal. This answers in the negative a question of Erd\H{o}s and Graham.

Locations

• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - View - PDF