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This question already has an answer here:

Is anything known about this sum?

$$ \sum_{\text{prime}\;p} \frac{1}{2^p} $$

I've calculated that it converges, but I can't determine whether the following related sum can be analytically continued outside the unit circle:

$$\sum_{\text{prime}\;p} z^{p}$$

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marked as duplicate by Xander Henderson, hardmath, Feng Shao, José Carlos Santos complex-analysis Sep 2 at 1:13

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    $\begingroup$ it blows up as $z \to 1$.... $\endgroup$ – mathworker21 Aug 24 at 16:18
  • $\begingroup$ indeed mathworker $\endgroup$ – Ultradark Aug 24 at 16:19
  • $\begingroup$ I guess you are asking if the sum is a lacunary function The primes don't satisfy the condition in the theorem of Hadamard given on the wiki, but it doesn't say that's a necessary condition. Maybe looking at the proofs of some of the theorems on lacunary functions will help. $\endgroup$ – saulspatz Aug 24 at 16:32
  • $\begingroup$ It looks like this immediately satisfies the conditions of Fabry’s extension to Hadamard’s theorem: en.m.wikipedia.org/wiki/Fabry_gap_theorem $\endgroup$ – Erick Wong Aug 24 at 16:54
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I will only address the second part of the question about $\sum\limits_{p \text{ prime}} z^p$.

There is a result by Pólya and Carlson.

Given any power series $f(z) = \sum\limits_{k=0}^\infty a_k z^k$ with integer coefficients and radius of convergence $1$, then $f(z)$ is either a rational function or has the unit circle $|z| = 1$ as natural boundary.

For the series at hand

$$f(z) = \sum\limits_{p \text{ prime}} z^p = \sum_{k=0}^\infty a_k z^k \quad\text{ where }\quad a_k = \begin{cases} 1, & p \text{ is prime}\\ 0, & \text{ otherwise }\end{cases}$$ It is trivial to see its radius of convergence is $1$. If $f(z)$ is rational, let's say $f(z) = \frac{p(z)}{q(z)}$ for polynomials $p(z)$ and $q(z)$, then $f(z)q(z) = p(z)$ is also polynomial.

WOLOG, assume $q(z)$ has the form $$q(z) = 1 + b_1 z + \cdots + b_n z^n \quad\text{ where } n = \deg(q)$$ Substitute this into the expression $f(z)q(z) = p(z)$ and compare coefficients of $z^{k+n}$ on both sides, we obtain

$$a_{k+n} + b_1 a_{k+n-1} + \cdots + b_n a_k = 0,\quad \text{ whenever }\quad k+n > \deg(p)$$

A consequence of this is if $a_k$ has infinitely many non-zero terms, then aside from initial $\deg(p)$ gaps, the gap between any two non-zero terms is at most $n-1$. It is known that there are infinitely many prime numbers and the prime gaps can be as large as one likes. This implies $f(z)$ is not rational.

By Pólya and Carlson, $\sum\limits_{p \text{ prime}} z^p$ has the unit circle as natural boundary. In other words, it cannot be analytic continued outside the unit disk in any manner.

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The main thing to know is for $x > 0$ and by analytic continuation for $\Re(x) > 0$ $$f(x) = \sum_{p^k} \log(p) e^{-x p^k}, \qquad \Gamma(s) \frac{-\zeta'(s)}{\zeta(s)} = \int_0^\infty f(x)x^{s-1}dx$$

$$f(x) = \frac1{2i\pi} \int_{2-i\infty}^{2+i\infty}\Gamma(s) \frac{-\zeta'(s)}{\zeta(s)} x^{-s}ds = \sum Res(\Gamma(s) \frac{-\zeta'(s)}{\zeta(s)} x^{-s})$$ $$ = x^{-1}- \sum_\rho \Gamma(\rho) x^{-\rho}+\sum_{k=0}^\infty x^k(a_k+b_k \log(x))$$

$f$ is very interesting mainly because $f(x+2 i \pi \frac{a}{q})$ has a similar explicit formula in term of $\frac{L'(s,\chi)}{L(s,\chi)}$ for the Dirichlet characters modulo $q$, thus it encodes the generalized Riemann hypothesis.

$\sum_{p^k}\frac1k e^{-xp^k},\sum_p e^{-xp}$ work the same way except $\Gamma(s) \log\zeta(s),\Gamma(s)P(s)$ aren't meromorphic so their explicit formulas are more complicated.

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  • $\begingroup$ @reuns can you explain this better? Maybe tailor your answer to my math level? $\endgroup$ – Ultradark Aug 24 at 16:30
  • $\begingroup$ what do you not understand ? those functions have explicit formulas in term of the non-trivial zeros of $\zeta(s)$, this is of course a complicated subject, but the point is to understand that everything we know of those functions come from their explicit formula, thus we concentrate on $f$ whose explicit formula is easier to define. From that we know almost everything : analytic continuation, asymptotic... Which depends on the (G)RH. I don't think we know any closed-form/special value for $f(x)$ nor $\sum_p e^{-xp}$. $\endgroup$ – reuns Aug 24 at 16:32
  • $\begingroup$ okay I understand better now :) $\endgroup$ – Ultradark Aug 24 at 16:37

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