Suppose $\alpha > 1$ is irrational. Are there infinitely many primes of the form $\left\lfloor \alpha n \right\rfloor$? Is the number of $p \leq X$ of this form $\sim \alpha^{-1} X (\log{X})^{-1}$? I know this is the kind of thing the circle method was born to do, but I cannot for the life of me find a reference for this!



Primes in quasi-arithmetic progressions?
I think the uniform distribution mod1 of $\{p/\alpha\}$ is due to Vinogradov, and the asmptotic for primes in a Beatty sequence $\sim \frac{\pi(x)}{\alpha}$ is an immediate consequence. Indeed for $p$ to be equal to some $\lfloor k\alpha\rfloor$ it is equivalent to $1-\frac{1}{\alpha}<\frac{p}{\alpha}-\lfloor \frac{p}{\alpha}\rfloor<1$. So you just need the fractional part of $p/\alpha$ to be on a fixed interval of length $\alpha$ mod1.


On a related note [this](http://arxiv.org/abs/0708.1015) paper discusses the general sequence $q\lfloor \alpha n+\beta\rfloor +a$.


