Electronic Communications in Probability

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

Paul Bourgade, Takahiko Fujita, and Marc Yor

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Abstract

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.

Article information

Source
Electron. Commun. Probab. Volume 12 (2007), paper no. 9, 73-80.

Dates
Accepted: 7 April 2007
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465224952

Digital Object Identifier
doi:10.1214/ECP.v12-1244

Mathematical Reviews number (MathSciNet)
MR2300217

Zentralblatt MATH identifier
1129.60088

Keywords
Cauchy variables stable variables planar Brownian motion Euler numbers

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Bourgade, Paul; Fujita, Takahiko; Yor, Marc. Euler's formulae for $\zeta(2n)$ and products of Cauchy variables. Electron. Commun. Probab. 12 (2007), paper no. 9, 73--80. doi:10.1214/ECP.v12-1244. https://projecteuclid.org/euclid.ecp/1465224952.


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