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With $p_n = n^{th}$ prime and $f(s):=\sum_{n=1}^\infty 1/p_n^s$ when the series converges. What is the status of the following questions: What is the abscissa of convergence of $f$? What are the values of $f$ at integers > 1?

user88693
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  • For $s$ complex. – user88693 Aug 02 '13 at 06:34
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    (I just corrected a typo in the original version: replaced "$p_n^s$" with "$1/p_n^s$".) – user88693 Aug 02 '13 at 06:37
  • al-Hwarizmi's "hint" (since deleted) now applies: $f$ is the Prime zeta function. The reference to Wikipedia supplied the abscissa of convergence, but no closed form evaluations at integers... – user88693 Aug 02 '13 at 06:42
  • I don't know if there's a good reason to believe that there are closed form expressions for $f(n), n \in \bf{Z}$. Just wondering. Also wondering about the zeros. (There's a plot of some zeros on the Wolfram Mathworld page.) The literature on this object seems pretty thin. – user88693 Aug 02 '13 at 19:36
  • Also see. Anyone care to explain the downvotes? – anon Aug 03 '13 at 08:08

2 Answers2

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There is indeed very limited literature on this. And the reason is that such is not been achieved yet. The key classical article on the prime zeta function is that of Fröberg. Then what could intuitively help you to build a broader understanding of the prime zeta function, also in relation to the Riemann zeta function, is a smart lecture over here. What indeed will help you rather understand the "why not" can be intuitively gathered from the relation between the prime zeta function, and "the Riemann zeta function, its derivative and the Möbius function".

In advance to the question "What is the abscissa of convergence?" one must find to a reasonable mathematical foundation, at least a rationale for a conjecture why and when "such abscissa shall be existent"? Then your second question of closed values at integers could follow.

In particular and for instance take into account that, (1) at all squarefree positive integers $\alpha$ we obtain singualrities in the form of bifurcation when either $s=1/\alpha$ or $s=\rho/\alpha$; while $\rho$ a non-trivial zero of the Riemann zeta function.

and (2) that we can gain: $$\sum_{n=1}^\infty \frac{P(ns)}{n}= \log \zeta(s)$$

while $P$ the prime zeta function.

General reference to wikki (and more extended) holds as requested by commentator of this answer.

al-Hwarizmi
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    When you say "such [has] not been achieved yet," you should specify you are referring to whether or not we know prime zeta at integer values, and conversely state clearly that the abscissa of convergence is well known to be ${\rm Re}(s)>1$, also with natural boundary ${\rm Re}(s)=0$ as an obstruction to analytic continuation (all stated on Wikipedia). – anon Aug 03 '13 at 08:06
  • yes. that was already communicated in my first comment that caught by the author of the question. I later deleted the reference to wikki as it is quite common. The "such" indeed referes to beyond that former comment. In any case you are correct. – al-Hwarizmi Aug 03 '13 at 08:28
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This is the prime zeta function, see e.g. http://en.wikipedia.org/wiki/Prime_zeta_function. The series converges for $\Re(s) > 1 $. On the Wiki page you can find the some function values for integer arguments.

gammatester
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