Difference between revisions of "Apéry's constant"
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(Created page with "Apéry's constant is the value $\zeta(3)$, where $\zeta$ denotes the Riemann zeta function. This constant is notable because it is known in general that for integers $n$,...") |
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| − | Apéry's constant is the value $\zeta(3)$ | + | 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 notoriously 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 05:55, 23 September 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 notoriously 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: █
