What is $$ \limsup\_{n \to \infty} \frac{\log(\mathrm{lcm}(1,2, \dots, n))}{n} \ \ ?$$



Asymptotics of the least common multiple of the first natural numbers
It is well-known that $\operatorname{lcm}(1,\ldots,n) = e^{\psi(n)}$, where $\psi$ is the Chebyshev's function. Since $\psi(x) = x + o(x)$, as $x \to +\infty$, (a form of the Prime Number Theorem) it follows that actually $\lim\_{n \to +\infty} \frac{\log(\operatorname{lcm}(1,\ldots,n))}{n} = 1$. (See Part 1 of G. Tenenbaum - Introduction to Analytic and Probabilistic Number Theory).


Not really a MO question, in my opinion.


