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 notoriously open problem to find $\zeta$ at odd integers. One of the first results in this area is the following theorem by Apéry.
+
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.
  
 
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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:

References

An Elementary Proof of of Apéry's Theorem