An inequality involving Euler’s $\varphi$-function

Question

Let $\varphi$ be Euler’s phi-function. I have seen it claimed that $$\varphi(n)/n = O(\log \log n).$$

Could someone either give a proof of this fact or tell me a reference where I can find this proof?

Answer 1

Since trivially $\phi(n)<n$ I suspect that it is about $\frac{n}{\phi(n)}=O(\log(\log(n))$. In Tenenbaum's Introduction to Analytic and Probabilistic Number Theory, page 84, Theorem 4 the following estimate is proved (in a paper by Rosser and Schönfeld) $$ \frac{n}{\phi(n)} < e^{\gamma} \log \log n + \frac{5}{2 \log\log n} $$ for all $n>223092870$.

Answer 2

this was the first elementary equivalent of the Riemann hypothesis; this is just the first page