# Geometric-like Sum over Primes

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.

• The answer to your second question is pretty obvious. The primes are unbounded. – Yves Daoust May 22 at 12:39
• Thank you @YvesDaoust I don't know how I missed that. – Brian May 22 at 13:05

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 $$\frac{1}{R}=\limsup_{n\rightarrow\infty}|a_n|^{1/n}=\lim_{n\rightarrow\infty} 1^{1/n}=1.$$

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.

• What is the connection with the question ? – Yves Daoust May 22 at 12:41