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Is there a known way to evaluate sums of the form

$$\sum_{p\text{ prime}} x^{p},$$ and are there any restrictions on the value of $x$ (e.g., $|x|<1$ for typical geometric series)?

EDIT: The second question was extremely obvious. Of course $|x|<1$ is required. I honestly don't know how I missed that.

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    $\begingroup$ The answer to your second question is pretty obvious. The primes are unbounded. $\endgroup$ – Yves Daoust May 22 at 12:39
  • $\begingroup$ Thank you @YvesDaoust I don't know how I missed that. $\endgroup$ – Brian May 22 at 13:05
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The restriction can be seen in one or two ways. The first is comparison to the geometric series. The second is that the series can be defined as $\sum\limits_{n=0}^\infty a_nz^n,$ where $a_n=1$ if $n$ is prime and $0$ otherwise. Then, the radius of convergence is defined as $$\frac{1}{R}=\limsup_{n\rightarrow\infty}|a_n|^{1/n}=\lim_{n\rightarrow\infty} 1^{1/n}=1.$$

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Partial answer. If $N \ge 2, x \ge 2$ then

$$ \lim_{N \to \infty} \sup \frac{\sum_{n \le N} x^n}{\sum_{p \le N} x^p} = \frac{x}{x-1} $$ which gives $$ {\sum_{p \le N} x^p} \approx (1-1/x){\sum_{n \le N} x^n} $$

converting the problem from a power series over primes to a much more managable power series over natural numbers.

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    $\begingroup$ What is the connection with the question ? $\endgroup$ – Yves Daoust May 22 at 12:41

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