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?

Thanks in advance.

statementof mathoverflow.net/questions/131018 (Vinogradov proved in 1935 that the fractional parts are not just dense but equidistributed). $\endgroup$ – Noam D. Elkies Dec 18 '13 at 22:28