Conjecture about the Prime Zeta Function

[David Carroll]

I was fooling around with the Prime Zeta Function just recently.

Prime Zeta Function, P(s), is defined as Σ1/(p^s), where p is each successive prime. When inputting various positive integer values for (s) on wolfram alpha, I noticed a pattern. Well, an approximate pattern, I should say.

My conjecture is that P(s) ≈ e/(the (s)th prime times the (s - 1)th prime).

E.g. (values computed on wolfram alpha)

P(2) ≈ .4522474 ≈ e/6 = e/(2 * 3) = e/(1st prime times the 2nd prime)
P(3) ≈ .1747626 ≈ e/15 = e/(3 * 5) = e/(2nd prime times the 3rd prime)
P(4) ≈ .0769931 ≈ e/35 = e/(5 * 7) = e/(3rd prime times the 4th prime)
P(5) ≈ .035755 ≈ e/77 = e/(7 * 11) = e/(4th prime times the 5th prime)

etc.

If any of this is correct (and if we could get rid of the errors), then we could divide P(s-1) by P(s) and get the ratio [(s + 1)th prime]/[(s - 1)th prime], but then I wouldn't know how to manipulate this any further to extract simply the (s + 1)th prime, the (s)th prime and so forth. But if we/I could, then we can input non-integer values into P(s) and define what the "2.5th" prime is, for example.

Any thoughts?

 

[David Carroll]

2 words: Never Mind.