0
$\begingroup$

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?

New contributor
user138371 is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
$\endgroup$
3
$\begingroup$

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

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

Your Answer

user138371 is a new contributor. Be nice, and check out our Code of Conduct.

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.