In a previous question Asymptotic growth of l.c.m. of all integers below $k$, it was noted that using the Prime Number Theorem you can prove that
$$ \log(\mbox{ lcm }(1,2,\ldots,k)) =k+\mbox{ o}(k)$$
Examining the values of $ \log(\mbox{ lcm }(1,2,\ldots,k))/k$, one finds that
They take an upward jump at each prime and to a lesser extent at each prime power.
They are near to $1$ once $k$ goes above $4$, but are usually just below $1$.
The occasional excursions above $1$ occur after high concentrations of primes, for example, when twin primes (or a prime gap of $4$ immediately following a twin prime) force the log to grow faster than the increase in $k$ can push it down.
Certain values of $k$ produce values of $ \log(\mbox{ lcm }(1,2,\ldots,k))/k$ that are higher than that for any lower $k$. The sequence of such values starts with $$ 2,3,4,5,7,9,11,13,19,31,109,113 $$
I am interested in two related properties:
Is there some finite value $\eta$ $$\eta = \sup\left(\frac{\log(\mbox{ lcm }(1,2,\ldots,k))}{k}\right)$$ for $k\in \Bbb{Z}, k>1$
such that for all $k$, $ \log(\mbox{ lcm }(1,2,\ldots,k))/k \leq \eta$?
And if there is such a bound, does the equality hold (in which case the sequence of "best $k$ thus far" is finite) or does the best value of $ \log(\mbox{ lcm }(1,2,\ldots,k))/k$ up to $k_0$ approach $\eta$ as $k_0 \to \infty$ but never get there?