Difference between revisions of "Riemann zeta"
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*[https://www.youtube.com/playlist?list=PL32446FDD4DA932C9 15 Videos about the Riemann $\zeta$ function] | *[https://www.youtube.com/playlist?list=PL32446FDD4DA932C9 15 Videos about the Riemann $\zeta$ function] | ||
| − | [http://www.claymath.org/sites/default/files/ezeta.pdf English translation of Riemann's paper "On the number of prime numbers less than a given quantity"] | + | *[http://www.claymath.org/sites/default/files/ezeta.pdf English translation of Riemann's paper "On the number of prime numbers less than a given quantity"] |
Revision as of 00:23, 19 October 2014
Consider the function $\zeta$ defined by the series $$\zeta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{n^z}.$$
Properties
Proposition: If $\mathrm{Re} \hspace{2pt} z > 1$, then the series defining $\zeta(z)$ converges.
Proof: █
Proposition (Euler Product): $\zeta(z)=\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z} = \displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}}$
Proof: █
