I'm currently reading the presentation of Selberg's sieve by Gelfond & Linnik, Elementary methods in the elementary theory of numbers. I have difficulties in evaluating the sum $$\sum_{d \leq \sqrt{z}}\frac{\mu(n)^2}{f_1(n)}$$ where $z = O(N)$, $N$ is an even integer, $$f_1(n) = \frac{n}{\tau_N(n)}\prod_{p \mid n}\left(1 - \frac{\tau_N(p)}{p}\right)$$ where $p$ denotes a prime and $$\tau_N(d) = \tau\left(\frac{d}{(d,N)}\right)$$ More details are given in the excerpt of the book in question (the computation of the sum begins at the bottom of page 124).
The problem is I fail to follow the calculations from the middle of page 125, when you're supposed to use the result:
$$\sum_{n \leq y}\frac{\tau(n)}{n} = \frac{1}{2}\ln(y)^2 + O(\ln(y))$$
As the authors have suggested, I tried to "inver[t] the order of summation" in all possible ways but couldn't figure out how to come to the result.
You will notice that the book has quite a few typos, making it a bit difficult to follow, but up to the point where I am stuck at, they are quite easy to correct.