Difference between revisions of "Euler product for Riemann zeta"
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| − | <strong>[[Euler product for Riemann zeta|Theorem]] (Euler Product):</strong> The following formula holds: | + | <strong>[[Euler product for Riemann zeta|Theorem]] (Euler Product):</strong> The following formula holds for $\mathrm{Re}(z)>1$: |
$$\zeta(z)=\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z} = \displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}},$$ | $$\zeta(z)=\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z} = \displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}},$$ | ||
where $\zeta$ is the [[Riemann zeta function]]. | where $\zeta$ is the [[Riemann zeta function]]. | ||
Revision as of 10:20, 30 December 2015
Theorem (Euler Product): The following formula holds for $\mathrm{Re}(z)>1$: $$\zeta(z)=\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z} = \displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}},$$ where $\zeta$ is the Riemann zeta function.
Proof: █
