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: | ||
| − | $$\zeta(z)=\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z} = \displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}} | + | $$\zeta(z)=\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z} = \displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}},$$ |
| + | where $\zeta$ is the [[Riemann zeta function]]. | ||
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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Revision as of 05:12, 4 September 2015
Theorem (Euler Product): The following formula holds: $$\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: █
