Good bound for $\sum_{k\leq X} \phi(k)/k$

Question

I am looking for a good bound the sum $\displaystyle S= \sum_{k\leq X} \phi(k)/k$, perhaps better than the trivial $O(X)$.

https://mathoverflow.net/a/84574/147650 does show that $\sum_{n<X}\phi(n)= \frac{1}{2\zeta(2)}X^2+O(\log X)$. Using this fact and by partial summation, we have that $S=\frac{1}{X}\sum_{k\leq X}\phi(k)-\int_1^X(\sum_{k\leq t}\phi(k))(-1/t^2)\mathrm{d}t$ which does not beat the trivial bound. Can we do better than $O(X)$?

Answer 1

We have $$\sum_{n\le x} \frac{\varphi(n)}{n} = \frac{6}{\pi^2} x + O(\log x)$$

This can be proved using $$\sum_{n\le x} \frac{\varphi(n)}{n} = \sum_{n\le x} \frac{\mu(n)}{n}\left\lfloor\frac{x}{n}\right\rfloor$$