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?

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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 $\Re(s) \ge 1, s \ne 1$.

  • $\begingroup$ That is not the series the OP asked about. $\endgroup$ – A. Bailleul 2 days ago

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