20090528, 17:30  #1 
Aug 2006
13533_{8} Posts 
Finite prime zeta?
Is there a good estimate for
? This is a generalization of the prime zeta function, as I'm particularly interested in n around 10^6 and s around 2, if it helps. And yes, I mean high precision. Saying for large n isn't helpful since I'm calculating (the highorder terms of P(s)). 
20090528, 18:43  #2 
Aug 2006
3×1,993 Posts 
If nothing else I can do
which Mathematica can probably integrate. But this doesn't seem to give good results. Last fiddled with by CRGreathouse on 20090528 at 18:43 
20090528, 22:05  #3  
Nov 2003
2^{2}·5·373 Posts 
Quote:
Try it as a Stieltje's integral. Integrate with respect to d pi(x) instead of dx. Use repeated integration by parts along with pi(x) ~ li(x). I don't know how this will turn out. I've never worked with this version of the zeta function. You might also try estimating the function itself using EulerMaclauren summation with respect to d pi(x) instead of dx. 

20090529, 20:38  #4 
Aug 2006
3·1,993 Posts 
Hmm, not a bad idea. Actually I think it would be best to apply this to the original problem directly; the source of the error may be in the use of the approximation leading to my use of the (truncated) prime zeta function. Perhaps not, but even so reducing error would be good.

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