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mathematical physics > special function > Riemann zeta function > Apéry's constant

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Terme préférentiel

Apéry's constant

In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number
${\\displaystyle {\egin{aligned}\\zeta (3)&=\\sum _{n=1}^{\\infty }{\ rac {1}{n^{3}}}\\\\&=\\lim _{n\ o \\infty }\\left({\ rac {1}{1^{3}}}+{\ rac {1}{2^{3}}}+\\cdots +{\ rac {1}{n^{3}}}\ ight),\\end{aligned}}}$
where $ζ$ is the Riemann zeta function. It has an approximate value of
$ζ (3) = 1.20205 6903159594285399738161511449990764986292 \dots$ (sequence A002117 in the OEIS).
The constant is named after Roger Apéry. It arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics. It also arises in the analysis of random minimum spanning trees and in conjunction with the gamma function when solving certain integrals involving exponential functions in a quotient, which appear occasionally in physics, for instance, when evaluating the two-dimensional case of the Debye model and the Stefan–Boltzmann law.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constant)

Apéry's theorem

constante d'Apéry

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