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i would like to know what is the exact relation between prime numbers and the zeta function. Thank you in advance !

Hptunjy Prjkeizg
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  • Have you heard of the Euler product formula? – manthanomen Jun 27 '17 at 05:18
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    see here http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Riffer-Reinert.pdf – Dr. Sonnhard Graubner Jun 27 '17 at 05:19
  • thank you ! i'll check it right now ! – Hptunjy Prjkeizg Jun 27 '17 at 05:21
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    $\zeta(s) = \sum_{n=1}^\infty n^{-s}$ is a very simple analytic function but $\frac{-\zeta'(s)}{\zeta(s)} = \sum_{p^k} p^{-sk} \log p$ is very complicated (and chaotic). By inverse Laplace transform we obtain that $|x-\sum_{p^k < x} \log p| < C x^{\sigma} \log^2 x$ iff $\frac{-\zeta'(s)}{\zeta(s)}-\frac{1}{s-1}$ is analytic (ie. $\zeta(s)$ has no zeros) on $\Re(s) > \sigma$. – reuns Jun 27 '17 at 10:02
  • Just as companion of previous comments and reference, I want to add an additional reference from the literature (I am aficionado, but this reference is very good). You can read the nice The lowest zeros of Riemann's zeta are in front of your eyes (October 30, 2014), from the blog of professor David Mumford. Good luck, isn't required a response of this comment. –  Jul 21 '17 at 20:17

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