Let $p_n$ be the $n$-th prime number, $n = 1, 2, \dots$

For which real values of $\mu$ does the Fourier series

$$\sum_{n=1}^\infty p_n^\mu e^{inx}$$

converge uniformly or absolutely in some non-trivial interval for $x$?

Are there any closed-form or alternative expressions for the sum when it does converge?