# Is $\{ p \alpha \}$ for prime $p$ dense in $[0,1]$?

Let $\alpha$ an irrational real number. It is well known that the set $\{ \{n \alpha \}|\,\, n \in \mathbb{N} \}$ is dense in$[0,1]$. ($\{x\}$ denotes the fractional part of $x$)

But how to prove the set $\{ \{p \alpha \}|\,\, prime\,\,p\}$ is dense in $[0,1]$? Also, is it uniformly distributed?

It is somehow a strange generalization of Dirichlet's theorem of primes on arithmetic progressions, so I tried to emulate its proof, by considering sums of the form:

$S(k)=\sum\limits_{prime\,\,p} \frac{f(k p\alpha)}{p}$

Where $f$ would be a function analogous to a character (then we can "isolate" the $p$'s such that $\{p\alpha\}$ is in some some interval $[a,b]$ by considering an adequate linear combination of the $S(k)$'s). However, I fail to bound the sums when simply choosing $f(x)=e^{2i\pi x}$, because it is not multiplicative as characters are. But, a construction of such function seems hopeless. Any ideas?

Dirichlet's theorem modulo $n$ is about proving $\{p/n\}$ is "uniformly distributed". This is like looking at the primes modulo $1/\alpha$. It is a big abstraction, but it gave me the idea of trying to solve the problem like that. – Rodrigo Dec 18 '13 at 22:36
Along these ideas, Vinogradov proved that $\{p\alpha\}$ is equidistributed in $[0,1]$ as $p$ runs through the prime numbers (in increasing order). Of course the rate of equidistribution depends on how well $\alpha$ can be approximated by rational numbers. For a newer version of this result see Corollary 2.2 in Vaughan: On the distribution of $\alpha p$ modulo 1, Mathematika 24 (1977), 135-141.