Prime numbers sinus density

[o.m.]

Let \( F=(\sin p\ ;\ p\in P) \), where \( P \) is the set of prime numbers.

Does F is dense in \( [-1;1] \) ?

[aleph]

Yes, because according to a theorem of Vinogradov, the sequence
\( \{a p : p \in P\} \)
is equidistributed modulo 1 so a fortiori dense modulo 1, for each irrational number \( a \).