I am reading A survey of gcd-sum functions where the following result is stated:
Let $P(n)$ be the Pillai's arithmetical function. The Dirichlet series of $P$ is given by: $$\sum_{n=1}^\infty \frac{P(n)}{n^s}= \frac{\zeta^2(s-1)}{\zeta(s)} $$ The convolution method applied to that equality leads to the asymptotic formulae: $$ \sum_{n\leq x}P(n) = \frac{1}{2\zeta(2)}x^2 \log x + \mathcal{O}(x^2)$$ $$ \sum_{n \leq x} \frac{P(n)}{n} = \frac{1}{\zeta(2)}x\log + \mathcal{O}(x)$$
My question is which method is the author referring to? I tried looking it up but haven't found something like that and the paper doesn't go too deep into that. Any help getting a reference that explains it in a greater extense? Thank you in advance!
The method used looks like the Dirichlet's Hyperbola Method, since we can have the asymptotic expansion of $\sum_{n\leq x}\phi(n)$ of the order $\frac{1}{2\zeta(2)}x^2$ and, loosely speaking, the theorem lets us write the partial sums of a dirichlet convolution between two functions in terms of the partial sums of the functions separately (keeping in mind that the referenced text shows that $P$ can be written as $P=n \ast \phi$, where $\phi$ is the Euler Totient function).
For reference, you can check:
Tenenbaum's book Introduction to Analytic and Probabilistic Number Theory, chapter of Average Orders;
Apostol's book Introduction to Analytic Number Theory, end of chapter 3.