Need reference to an assertion re: Riemann

Question

I have been exploring the patterns of the greatest divisors $\leq$ to the square root $n$ for sequences of size $n$. Just for grins, I used $\textit{Mathematica}$ to produce this ListLinePlot:

When I saw how symmetrical it was, I had to look up the sequences: OEIS A003418 The third sentence in the comments states that this sequence relates to this assertion regarding Riemann: for $n > 2$

$$|\log (\text{lcm}(n))-n |<\sqrt{n} \log ^2(n)$$

Where could I find a book or paper that describes that assertion?

Answer 1

We have $$ \log\text{lcm}(1,\dots,n)=\sum_p\log p\left[\frac{\log n}{\log p}\right] =\sum_p \log p\sum_{k\log p\leq \log n} 1 = \sum_{p^k\leq n}\log p.$$ The right hand side is the summatory function of the von Mangoldt function, which is known as the second Chebyshev function: $$ \psi(n):=\sum_{m\leq n}\Lambda(m). $$ That is, the bound you are asking about is $$ |\psi(n)-n|<\sqrt{n}\log^2 n.$$ The fact that the Riemann Hypothesis implies this bound for $n>73$ (hence probably also for $n>2$) follows from this paper. The other direction, that such a bound with any constant in front of $\sqrt{n}\log ^2 n$ implies the Riemann Hypothesis, is more classical and can be found in many textbooks (see e.g. the bottom of p.463 in Montgomery-Vaughan: Multiplicative Number Theory I).