I am doing some analytic number theory and I'm strugling with showing this result in particular:
$\sum_{n\leq x,(n,k)=1}\frac{1}{n} = \frac{\phi(x)\log(x)}{k} + O(1)$
Where $k$ is some constant. In general, I am unsure how to approach series with the coprime requirement in the summation index. I have tried Abel Summation, re working indices, the usual number theory tricks. Any tips would be appreciated!
Using $\sum_{d | n} \mu(d) = 1_{n=1}$, $\sum_{d | k}\phi(d)= k \implies \phi(k) = \sum_{d | k}\mu(d) \frac{k}{d}$ and $\sum_{n \le x} \frac{1}{n}=\log(x)+\gamma+\mathcal{O}(1/x)$ we obtain $$\sum_{n \le x, (n,k)=1} \frac{1}{n} = \sum_{d | k} \mu(d) \sum_{n \le x, d | (n,k)} \frac{1}{n} = \sum_{d | k} \mu(d) \sum_{n \le x/d} \frac{1}{nd}$$ $$=\sum_{d | k} \frac{\mu(d)}{d} (\log x-\log d+\gamma+\mathcal{O}(d/x))= \frac{\phi(k)}{k}(\log x+\gamma)-\sum_{d | k} \frac{\mu(d)}{d} \log d+\mathcal{O}(\frac{\sigma_0(k)}{x})$$