The number of relatively coprime integers less than $N$ grows like $\frac{N^{2}}{\zeta(2)}$ (which, looking at the structure of $\frac{1}{\zeta(2)}=\Pi_{p_{k}}(1-\frac{1}{p_{k}^{2}})$, can be proven through an approximate inclusion exclusion argument which I don't understand how to truncate satisfactorily and which ties with the next question).
How do we get the best estimates for the error for this count? Noam Elkies' lectures in Analytic Number theory asks to prove an estimate of $O(N log N)$.
Edit: Thanks to reuns for the answer, and the calculation using the mobius function.
$$2\sum_{n = 1}^N \phi(n) =2\sum_{n=1}^N \sum_{d | n}\mu(d) \frac{n}{d} =2 \sum_{d=1}^N \mu(d) \sum_{m=1}^{ \lfloor N/d\rfloor} m= \sum_{d=1}^N \mu(d) \lfloor N/d\rfloor(\lfloor N/d\rfloor+1)$$ $$=\sum_{d=1}^N \mu(d) (N^2/d^2+O(N/d)) = N^2(\frac{1}{\zeta(2)}+O(1/N)) + O(N \log N)$$ Also $2\sum_{n=1}^\infty n^{-s} \phi(n) = 2\frac{\zeta(s-1)}{\zeta(s)}$