Euler's formulae for $\zeta(2n)$ and products of Cauchy variables

Paul Bourgade, Takahiko Fujita, Marc Yor

doi:10.1214/ecp.v12-1244

Euler's formulae for $\zeta(2n)$ and products of Cauchy variables

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Paul Bourgade

Laboratoire de probabilités et modèles aléatoires, université Paris 6
Takahiko Fujita

Graduate School of Commerce and management, Hitotsubashi University
Marc Yor

Laboratoire de probabilités et modèles aléatoires, université Paris 6

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説明

We show how to recover Euler's formula for $\zeta(2n)$, as well as $L_{\chi_4}(2n+1)$, for any integer $n$, from the knowledge of the density of the product $\mathbb{C}_1,\mathbb{C}_2\ldots,\mathbb{C}_k$, for any $k\geq 1$, where the $\mathbb{C}_i$'s are independent standard Cauchy variables.