Density of primes of the form $kn\pm r$ If I'm not mistaken, Dirichlet's theorem states that if $(k,r)=1$ and $r<k$ then $\sharp\{p=kn+r,\mathbb{P}\ni p\leq x\}\sim \sharp\{q=kn-r,\mathbb{P}\ni q\leq x\}\sim\dfrac{1}{\varphi(k)}\pi(x)$.
My first question is: is this true?
If so, can we deduce from this theorem that for almost all positive $r$ (i.e a set of asymptotic density $1$),  $$\lim_{x\to\infty}G_{r}(x):=\sharp\{n\leq x,\vert (n,r)=1, r<n, (n-r,n+r)\in\mathbb{P}^{2}\}=\infty$$ ?
Many thanks in advance.
 A: Up to a constant, the function $G_r(x)$ measures the number of prime pairs at distance $2r$ from each other. It is a well-known conjecture stating that this function tends to infinity for every $r$ as $x$ grows without bound. However, the results on bounded intervals containing primes (see here) can be used to show that for at least (roughly, up to a multiplicative constant I think) $\log n$ numbers $r$ below $n$ this function grows without bound.
The results regarding bounded prime intervals are much more complicated than asymptotic version of Dirichlet's theorem on arithmetic progressions. There is no known way to deduce anything regarding the former using the latter, for a simple reason that it tells us nothing about how primes in different arithmetic progressions relate to each other. It would be consistent with Dirichlet theorem that for no number $r$ there are infinitely many prime pairs at distance $2r$ from each other (one can show this by constructing a subset of $\Bbb N$ which satisfies analogue of Dirichlet's theorem, but has gaps growing without bound. I can't give you details of such a construction, but it's easy to convince yourself that such a set exists).
