Euler’s formulae for ζ(2n) and products of cauchy variables

Paul Bourgade; Takahiko Fujita; Marc Yor

doi:10.1214/ECP.v12-1244

Euler’s formulae for ζ(2n) and products of cauchy variables

Paul Bourgade, Takahiko Fujita, Marc Yor

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We show how to recover Euler’s formula for ζ(2n), as well as L_χ4 (2n + 1), for any integer n, from the knowledge of the density of the product ℂ₁,ℂ₂., ℂ_k, for any k ≥ 1, where the ℂ_i’s are independent standard Cauchy variables.

Original language	English (US)
Pages (from-to)	73-80
Number of pages	8
Journal	Electronic Communications in Probability
Volume	12
DOIs	https://doi.org/10.1214/ECP.v12-1244
State	Published - Jan 1 2007

Keywords

Cauchy variables
Euler numbers
Planar Brownian motion
Stable variables

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1214/ECP.v12-1244

Cite this

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abstract = "We show how to recover Euler{\textquoteright}s formula for ζ(2n), as well as Lχ4 (2n + 1), for any integer n, from the knowledge of the density of the product ℂ1,ℂ2., ℂk, for any k ≥ 1, where the ℂi{\textquoteright}s are independent standard Cauchy variables.",

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author = "Paul Bourgade and Takahiko Fujita and Marc Yor",

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AU - Bourgade, Paul

AU - Fujita, Takahiko

AU - Yor, Marc

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N2 - We show how to recover Euler’s formula for ζ(2n), as well as Lχ4 (2n + 1), for any integer n, from the knowledge of the density of the product ℂ1,ℂ2., ℂk, for any k ≥ 1, where the ℂi’s are independent standard Cauchy variables.

AB - We show how to recover Euler’s formula for ζ(2n), as well as Lχ4 (2n + 1), for any integer n, from the knowledge of the density of the product ℂ1,ℂ2., ℂk, for any k ≥ 1, where the ℂi’s are independent standard Cauchy variables.

KW - Cauchy variables

KW - Euler numbers

KW - Planar Brownian motion

KW - Stable variables

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JO - Electronic Communications in Probability

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ER -

Euler’s formulae for ζ(2n) and products of cauchy variables

Abstract

Keywords

ASJC Scopus subject areas

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