Difference between revisions of "Apéry's constant"

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Apéry's constant is the value
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Apéry's constant is the number
$\zeta(3)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k^3},$
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$\zeta(3)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k^3}$,
 
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.
 
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=
 
=Properties=
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[[Apéry's constant is irrational]]<br />
<strong>Theorem:</strong> The following formula holds:
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[[Relationship between integral of x*log(sin(x)), and Apéry's constant, pi, and logarithm]]
$$\displaystyle\int_0^{\frac{\pi}{2}} x \log(\sin(x)) dx = \dfrac{7}{16}\zeta(3) - \dfrac{\pi^2}{8} \log(2).$$
 
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<strong>Proof:</strong> █
 
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<strong>Theorem: (Apéry)</strong> The number $\zeta(3)$ is irrational.
 
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<strong>Proof:</strong> █
 
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=References=
 
=References=
 
[http://arxiv.org/pdf/math/0202159v1.pdf An Elementary Proof of of Apéry's Theorem]
 
[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