Are sines of primes dense in $[-1,1]?$

Solution 1:

According to the Wikipedia article about the discrepancy of a sequence:

The sequence of all multiples of an irrational $\alpha$ by successive prime numbers, $2 \alpha$, $3 \alpha$, $5 \alpha$, $7 \alpha$, $11 \alpha$, ... is equidistributed modulo 1. This is a famous theorem of analytic number theory, proved by I. M. Vinogradov in 1935.

With $\alpha = \frac{1}{2 \pi}$, this implies that $P$ is equidistributed modulo $2 \pi$. Using this, and the continuity of the sine function, I think it is straightforward to show that $\sin(P)$ is dense in $[-1,1]$ (although not equidistributed).