Dirichlet's theorem possible generalization

Question

Looking at the generalizations of Dirichlet's theorem on arithmetic progressions, I have not seen one that I thought about recently:

Possible generalization

Let $a,k$ be positive coprime integers, and $S$ some set of positive integers such that:

(i) $\forall x\in S$, $\gcd\left(a,x\right)=\gcd\left(k,x\right)=1$

(ii) $\sum_{x\in S} x^{-1}=\infty$

Then, there are infinitely many prime numbers of the form $ax+k$.

This possible generalization seems sound with some examples I have thought about. For instance:

• $a=2$,$k=1$, $S=\{x:2n-1, n\in \mathbb N\}$ (the sequence $ax+k$ contains all the primes of the form $4k+3$, already proved there are infinitely many).
• $a=2$,$k=1$, $S=\{x:p, p\in \mathbb P\}$ (infinitely many Sophie Germain primes conjecture).

I would like to know if there is some recent paper related to the conjecture, and assessment about how difficult could it be to prove it, as well as some mathematical machinery that you would recommend me to study in order to try to attack it.

Thanks!

Answer 1

This conjecture is false. Some counterexamples:

• One can take $a=2$ and $k=1$ again, but take $S=\{7,13,19,25,\dots\}$; then $ax+b$ is a multiple of $3$ for every $x\in S$.
• One can also take $a=k=1$ and take $S=\{3,5,7,8,9,11,13,14,15,\dots\}$ to be the set of all numbers $x$ such that $x+1$ is composite.