Density zero subsets of primes

Question

Let $P$ denotes the set of all primes. Find a subset $Q$ of $P$ such that the following holds:

(i) Relative density of $Q$ is zero in $P$ and

(ii) $ \sum_{p \in Q} \frac{1}{p} = \infty $.

Answer 1

One easy way to create sets of primes with "user-specified" density is the following theorem of Fouvry and Iwaniec ("Gaussian Primes," Fouvry, Iwaniec, 1997):

Let $\lambda_\ell$ be complex numbers with $|\lambda_\ell| \leq 1$. Then $$\sum_{m^2+\ell^2 \leq x} \lambda_\ell\Lambda(\ell^2+m^2) = \sum_{\ell^2+m^2 \leq x} \lambda_\ell \psi(\ell) + O_A \left(x (\log x)^{-A} \right),$$ where $\Lambda$ is the von Mangoldt function, $$\psi(\ell) = \prod_{p \ \nmid \ \ell} \left(1 - \frac{\chi(p)}{p-1} \right),$$ $\chi$ is the non-trivial character modulo 4, and $A$ is any fixed positive constant.

Now let us specialize. Let $Q$ be the set of primes of the form $p = m^2+\ell^2$, $\ell \geq 2017$, and $\ell$ divisible by $2^\alpha$ with $\alpha \geq \frac{\log \log \log \ell}{\log 2}$. Define $$\pi_Q(x) = \sum_{\substack{p \leq x \\ p \in Q}} 1.$$ Using the Fouvry-Iwaniec theorem one may show (exercise!) that $$ \pi_Q(x) \asymp \frac{x}{\log x \log \log x},$$ from which it follows easily by partial summation that $$\sum_{\substack{p \leq x \\ p \in Q}} \frac{1}{p} \gg \log \log \log x.$$