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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!

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Please reformulate the question in such a way that you don't ask for infinitely many primes p ≤ X. –  François G. Dorais May 21 '10 at 18:36
@FGD: Whoops! Done. –  David Hansen May 22 '10 at 18:08

2 Answers 2

up vote 8 down vote accepted

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 paper discusses the general sequence $q\lfloor \alpha n+\beta\rfloor +a$.

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"...is an immediate consequence." No, it's not, because n doesn't have to be prime for $\left \lfloor \alpha n \right \rfloor$ to be prime. –  David Hansen May 21 '10 at 17:45
Sorry for stating it like that, I added a clarification. –  Gjergji Zaimi May 21 '10 at 18:11
Oh, now I see! Thanks! –  David Hansen May 21 '10 at 19:11

Imre Ruzsa observed that since $p/\alpha$ is equidistributed modulo $1$, we have infinitely many primes $p$ for which the fractional part of $p/\alpha$ is less than $1/\alpha$. Writing $p/\alpha = n_p - {\epsilon}_p$ with $n_p$ an integer and $0 < {\epsilon}_p < 1/\alpha$, we get $p = {\alpha}n_p - {\alpha}{\epsilon}_p$ and thus ${\lfloor}{\alpha}n_p{\rfloor}$ prime for infinitely many distinct positive integers $n_p$.

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Yeah, I wasn't sure what the "local obstructions" for $\left\lfloor \alpha n \right \rfloor$ being prime would be exactly! –  David Hansen May 21 '10 at 16:24
I didn't quite remember the details correctly, but I have fixed them now. I would have to think more about the asymptotic estimate, so I have withdrawn my remarks on that. –  engelbrekt May 21 '10 at 16:34

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