I'm very confident that $$\sum_{p \ \text{prime}} \sin p $$

diverges. Of course, it suffices to show that there are arbitrarily large primes which are not in the set $\bigcup_{n \geq 1} (\pi n - \epsilon, \pi n + \epsilon)$ for sufficiently small $\epsilon$. More strongly, it seems that $\sin p$ for prime $p$ is dense in $[-1,1]$.

This problem doesn't seem that hard though. Here's something that (to me) seems harder.

If $p_n$ is the nth prime, what is $$\limsup_{n \to +\infty} \sum_{p \ \text{prime} \leq p_n} \sin p?$$ What is $$\sup_{n \in \mathbb{N}} \sum_{p \ \text{prime} \leq p_n} \sin p? $$

Of course, we can ask analogous questions for $\inf$.

I'm happy with partial answers or ideas. For example, merely an upper bound.