Finding an upper bound for LCM with conditions.

Question

Let $$\psi (x) = \sum_{\substack{p,m} \\ p^m \leq x} \log p.$$ I showed that $\psi (n) = \log \text{lcm}(1,\cdots,n)$ and $\psi (x) = O(x)$. How can I show that $\text{lcm}(1,\cdots,n) < A ^n$ for some constant $A$?

Answer 1

Looking at your question again, you know there is some $C$ with $\psi(x) < C x,$ so just take $A = e^C$

Theorem 12 in Rosser and Schoenfeld (1962): The quotient $\psi(x) / x$ takes its maximum at $x = 113,$ and $$ \psi(x) < 1.03883 \; x $$ so then take $e$ to both sides

Alright, there is a discussion of 1.03883 here: https://oeis.org/A206431 where more accuracy gives

1.038820577609129893008155562738246526933611208454503482505898