Difference between revisions of "Riemann zeta"
From specialfunctionswiki
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$$\zeta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{n^z}.$$ | $$\zeta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{n^z}.$$ | ||
| − | == | + | ==Properties== |
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
<strong>Proposition:</strong> If $\mathrm{Re} \hspace{2pt} z > 1$, then the series defining $\zeta(z)$ converges. | <strong>Proposition:</strong> If $\mathrm{Re} \hspace{2pt} z > 1$, then the series defining $\zeta(z)$ converges. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Proposition (Euler Product):</strong> $\zeta(z)=\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z} = \displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}}$ | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 00:45, 3 August 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: █
