There seems to be a lot of open questions about the lucky numbers, so let me phrase what we know about them and what we don't. Most of the questions are written as the equivalent forms of some known results for the prime numbers.
What we know
- The Lucky Numbers Theorem (equivalent form of the Prime Number Theorem for lucky numbers) - if we let $\pi_{s}(x)$ be the number of lucky numbers $\leqslant x$, then $$\lim_{x \to \infty}\frac{\pi_{s}(x) \log x}{x} = 1.$$ This result can actually be deduced from the asymptotic formula for the $n-$th lucky number $s_n$, given by S. Chowla here, $$ s_n = n\log n + \frac{n}{2}(\log \log n)^2 + o(n (\log \log n)^2).$$
What we don't know
Infinitude of the lucky prime numbers - are there infinitely many numbers that are both lucky and prime?
Twin lucky numbers - are there infinitely many numbers $s$ such that both $s$ and $s+2$ are lucky numbers?
The Goldbach conjecture for lucky numbers - does for any integer $n \geqslant 1$ the equation $$2n = s_1 + s_2$$ has a solution for $s_1, s_2$ being lucky?
Dirichlet's theorem for the lucky numbers - if we let $\pi_s(x;q,a)$ be the number of lucky numbers $s \leqslant x$, $s \equiv a (\mod{q})$, then how does $\pi_{s}(x;q,a)$ behave as $x$ tend to infinity?
Most of the problems associated with the lucky numbers are due to the fact that sieving process consists of iterations done not always on the whole set of all positive integers, but on the set given after the previous iteration. This makes everything a lot more complicated, as the Riemann zeta function and zeta-like functions are mostly suitable for the sets defined by iterations of sieving processes, all done on the same set.
In my little paper in polish (here) I have proven that, if we let $A_{k}$ be the set of all the integers sifted in the $k-$th iteration of the process (so every $s_k-$th of the remaining), then there exists a positive integer $q_k$ and some set $R_k \subset \{0,1,\ldots,q_{k}-1\}$ such that $$ A_{k} = \{iq_k + r \colon i \geqslant 0, r \in R_{k}\}$$ where $i \geqslant 0$ runs through nonnegative integers. I was looking for a theorem of that type as it could allow to use tools like the Dirichlet's theorem or related in a preferable form (like Brun-Titchmarsch inequality or Siegel-Walfisz theorem) to solve the problem of lucky primes. However, due to some complications in the sieving process at places where $q_k > x$, it still isn't enough. However, it does indeed show that with some (infinitely many, in fact) choices of $q$ and $a$ we'll have $\pi_{s}(x;q,a)=0$ for all $x>0$.