# Convergence of prime power Fourier series

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?

If $$\sum_p p^{-s} e^{2i \pi ap/q}$$ converges then so does $$\lim_{t \to 0}\sum_p p^{-s} e^{2i \pi ap/q}e^{-pt}$$
For $$\Re(s) < 1$$ the PNT in arithmetic progressions shows that for $$\gcd(a,q)=1$$ as $$t \to 0$$ for every $$k$$ $$\sum_p p^{-s} e^{2i \pi ap/q}e^{-pt} = \frac{\mu(q)}{\varphi(q)}\sum_{n=2}^\infty \frac{n^{-s}}{\ln n} e^{-nt}+o(\sum_{n=2}^\infty \frac{n^{-s}}{\ln^k n} e^{-nt})$$
Therefore $$\sum_p p^{-s} e^{i p x}$$ converges in some interval iff $$\Re(s) \ge 1, s \ne 1$$.