How can I prove: $\sum_{n\leq x}\frac{\phi(n)}{n^2} = \frac{\log x}{\zeta(2)}+\frac{C}{\zeta(2)} + A + O\left(\frac{\log x}{x}\right)$ The problem is that prove that $$\sum_{n\leq x}\frac{\phi(n)}{n^2} = \frac{\log x}{\zeta(2)}+\frac{C}{\zeta(2)} + A + O\left(\frac{\log x}{x}\right)$$ 
where $C$ is Euler's constant and $A = \sum_{n \geq 1}\frac{\mu(n)\log n}{n^2}$

The following is things I did try:
$$\sum_{n\leq x}\frac{\phi(n)}{n^2} = \sum_{n\leq x}\frac{1}{n}\frac{\phi(n)}{n} =\\ 
\sum_{n\leq x}\frac{1}{n}\sum_{d\mid n} \frac{\mu(d)}{d} = \sum_{q\leq x}\frac{1}{q} \sum_{d\leq x} \frac{x}{q}\frac{\phi(d)}{d^2} =\\
\sum_{q\leq x}\frac{1}{q}\left( \frac{1}{\zeta(2)} + O\left(\frac{q}{x}\right) \right) = \frac{\log x}{\zeta(2)} + \frac{C}{\zeta(2)}  + O\left(\frac{1}{x}\right) + \sum_{q\leq x}\frac{1}{q} O\left(\frac{q}{x}\right) $$
Here $\mu$ is the Möbius function and $\phi$ is the Euler totient function.

Let we implement the approach suggested by Winther in the comments, with a minor variation.
From
$$ \sum_{n\leq x}\varphi(n) = \frac{x^2}{2\zeta(2)}+O(x\log x) \tag{1}$$
and Abel's summation formula we get:
$$ \sum_{n\geq x}\frac{\varphi(n)}{n^2}=\frac{1}{2\zeta(2)}+O\left(\frac{\log x}{x}\right)+2\int_{1}^{x}\left( \frac{u^2}{2\zeta(2)}+O(u\log u)\right)\frac{du}{u^3}\tag{2} $$
and the claim readily follows by rearranging terms.
