Timeline for Convergence of the series $\sum_p p^{-s}$ ($p$ prime and $s>1$)
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Feb 22, 2015 at 19:58 | comment | added | Terry Tao | From the identity Peter mentioned and Mobius inversion one has $J(s) = \sum_{n=1}^\infty \frac{\mu(n)}{n} \log \zeta(ns)$. So for even $s$ one can get a series formula for $J$ this way, but it is unlikely to lead to any particularly compact closed form. | |
| Feb 22, 2015 at 19:16 | history | edited | Peter Humphries | CC BY-SA 3.0 |
fixed LaTeX and reverted to standard notations for sums over primes
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| Feb 22, 2015 at 19:02 | history | edited | Johannes Hahn | CC BY-SA 3.0 |
Added LaTeX
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| Jun 13, 2011 at 16:30 | history | edited | Charles | CC BY-SA 3.0 |
deleted 2 characters in body; edited tags
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| Jun 13, 2011 at 12:16 | answer | added | user9072 | timeline score: 5 | |
| Jun 13, 2011 at 12:12 | comment | added | Gerry Myerson | I believe $J(s)$ is sometimes called "the prime zeta function" and information about it can be found by using that search term. | |
| Jun 13, 2011 at 6:40 | comment | added | C.S. | This page: mathworld.wolfram.com/RiemannZetaFunction.html answers some for values for $\zeta(s)$ | |
| Jun 13, 2011 at 6:31 | comment | added | Peter Humphries | As an aside, one way of showing that the sum over primes diverges is via the identity $\sum_{p}{p^{-s}} = \log \zeta(s) - \sum_{p}\sum^{\infty}_{n=2}{n^{-1} p^{-ns}}$, which is valid for all $\Re(s) > 1$. This shows the connection between special values of $\sum_{p}{p^{-s}}$ and of $\zeta(s)$, but I don't think it's possible to obtain nice closed-form values for $\sum_{p}\sum^{\infty}_{n=2}{n^{-1} p^{-ns}}$ (though it is of course easy to show that it is uniformly bounded as $s \to 1$). | |
| Jun 13, 2011 at 6:26 | comment | added | C.S. | @Yemon: True, I misunderstood the question. I thought, he was asking formula known for $\zeta(s)$. | |
| Jun 13, 2011 at 6:09 | history | asked | bulai | CC BY-SA 3.0 |