I would like to define the most accurate as possible lower and upper bounds for $LCM([1..n])$, the least common multiple of a given interval $[1..n]$ (in old books is written as $[n]$).
Initially due to the fundamental theorem of arithmetic I just can imagine these very simple options:
Lower bound:
$$LCM([1..n]) \gt n\#$$
Where $n\#$ is the primorial of $n$, in other words, the product of the primes $p \le n$.
Upper bound:
$$LCM([1..n]) \le n!$$
The factorial must be always greater or equal than the LCM of the interval. So summarizing:
$$n\# \le LCM([1..n]) \le n!$$
They are easy to define, but they are not very accurate (closer to the value of the LCM).
I would like to ask the following questions:
Are the proposed bounds correct?
Are there more accurate lower and upper bounds? Thank you!
The power of $2$ that appears in the factorization of the least common multiple is the largest power of $2$ that is at most $n$. That is, $\lfloor \log_2 n \rfloor$. The power of $3$ that appears is similarly $\lfloor \log_3 n \rfloor$, and so on. So the actual lcm is exactly equal to
$$\prod_{p \le n} p^{\lfloor \log_p n\rfloor} = \prod_{p \le n} p^{\lfloor \log n / \log p\rfloor}$$ where the products are taken over primes. This product is exactly $\exp \psi(x)$ where $\psi(x)$ is the second Chebyshev function, for which lots of asymptotics are known.