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In the course of my work, I encountered the following sum ($(x,y)$ stands for the GCD of $x$ and $y$): $$L(Q)=\sum_{\substack{\delta_1,\delta_2\leq Q\\\delta_1\equiv0\ (a)\\\delta_2\equiv0\ (b)}}\frac{...

Take a set $\mathcal P$ of primes and denote by $\langle \mathcal P\rangle $ the set of all natural numbers composed of primes from $\mathcal P$. If \[ \sum _{p\in \mathcal P}\frac {1}{p}\] converges ...