Difference between revisions of "Apéry's constant"
From specialfunctionswiki
| Line 1: | Line 1: | ||
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 | + | 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. One of the first results in this area is the following theorem by Apéry. |
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
Revision as of 01:16, 19 October 2014
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. One of the first results in this area is the following theorem by Apéry.
Theorem: The number $\zeta(3)$ is irrational.
Proof: █
