Difference between revisions of "Apéry's constant"

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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 $n$, $\zeta(2n)$ 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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Apéry's constant is the number
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$\zeta(3)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k^3}$,
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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.
  
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=Properties=
<strong>Theorem:</strong> The number $\zeta(3)$ is irrational.
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[[Apéry's constant is irrational]]<br />
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[[Relationship between integral of x*log(sin(x)), and Apéry's constant, pi, and logarithm]]
<strong>Proof:</strong> █
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=References=
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[http://arxiv.org/pdf/math/0202159v1.pdf An Elementary Proof of of Apéry's Theorem]
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[[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

References

An Elementary Proof of of Apéry's Theorem