Question from the proof of the Prime Number Theorem

Question

My question is pretty trivial, but I just wanted to ask about something I can't see at all.

In the proof of the PNT supplied in these notes, it is asserted that when $|t| \ge 2$ $$O(\log^9(|t|)/|t|^2)=O(|t|^{-3/2})$$ on page 39. I cannot for the life of me see why. Any help would be appreciated.

Answer 1

It can be shown through l'Hopital's rule that $$\frac{\log^9(|t|)}{|t|^{1/2}} \to 0$$ as $|t| \to \infty$.

This means that $\log^9(|t|) = o(|t|^{1/2}) = O(|t|^{1/2})$. Hence $$\frac{\log^9(|t|)}{|t|^2} \le C \frac{|t|^{1/2}}{|t|^{2}} = C \frac{1}{|t|^{3/2}}.$$

Answer 2

We have $\log (|t|)=O(|t|^{1/18})$, so that $$ \frac{\log^9 (|t|)}{|t|^2}= O\left(\frac{|t|^{1/2}}{|t|^2}\right)=O(|t|^{-3/2}). $$