Difference between revisions of "Apéry's constant"
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| − | Apéry's constant is the | + | Apéry's constant is the number |
| − | $\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 | + | 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 "nice" closed formula exists for values $\zeta(2n+1)$. Hence it became a notorious open problem to find $\zeta$ at odd integers. |
| − | + | =Properties= | |
| − | < | + | [[Apéry's constant is irrational]]<br /> |
| − | + | [[Relationship between integral of x*log(sin(x)), and Apéry's constant, pi, and logarithm]] | |
| − | + | ||
| − | + | =References= | |
| − | + | [http://arxiv.org/pdf/math/0202159v1.pdf An Elementary Proof of of Apéry's Theorem] | |
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 17:17, 24 June 2016
Apéry's constant is the number $\zeta(3)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k^3}$, 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 "nice" closed formula exists for values $\zeta(2n+1)$. Hence it became a notorious open problem to find $\zeta$ at odd integers.
Properties
Apéry's constant is irrational
Relationship between integral of x*log(sin(x)), and Apéry's constant, pi, and logarithm
