Difference between revisions of "Apéry's constant"
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Apéry's constant is the value | Apéry's constant is the value | ||
$\zeta(3)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k^3},$ | $\zeta(3)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k^3},$ | ||
| − | where $\zeta$ denotes the [[Riemann zeta function]]. This constant is notable because it is known in general that for integers $n$, $\zeta(2n)$ is a rational multiple of $\pi$ but no general formula exists for values $\zeta(2n+1)$. Hence it became a notorious open problem to find $\zeta$ at odd integers | + | where $\zeta$ denotes the [[Riemann zeta function]]. This constant is notable because it is known in general that for integers $n$, $\zeta(2n)$ is a rational multiple of $\pi$ but no general formula exists for values $\zeta(2n+1)$. Hence it became a notorious open problem to find $\zeta$ at odd integers. |
| + | =Properties= | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| − | <strong>Theorem:</strong> The number $\zeta(3)$ is irrational. | + | <strong>Theorem:</strong> The following formula holds: |
| + | $$\displaystyle\int_0^{\frac{\pi}{2}} x \log(\sin(x)) dx = \dfrac{7}{16}\zeta(3) - \dfrac{\pi^2}{8} \log(2).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem: (Apéry)</strong> The number $\zeta(3)$ is irrational. | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 05:28, 12 April 2015
Apéry's constant is the value $\zeta(3)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k^3},$ where $\zeta$ denotes the Riemann zeta function. This constant is notable because it is known in general that for integers $n$, $\zeta(2n)$ is a rational multiple of $\pi$ but no general formula exists for values $\zeta(2n+1)$. Hence it became a notorious open problem to find $\zeta$ at odd integers.
Properties
Theorem: The following formula holds: $$\displaystyle\int_0^{\frac{\pi}{2}} x \log(\sin(x)) dx = \dfrac{7}{16}\zeta(3) - \dfrac{\pi^2}{8} \log(2).$$
Proof: █
Theorem: (Apéry) The number $\zeta(3)$ is irrational.
Proof: █
