Electronic Communications in Probability

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

Paul Bourgade, Takahiko Fujita, and Marc Yor

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Cauchy variables stable variables planar Brownian motion Euler numbers

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


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

Export citation


  • Andrews, George E.; Askey, Richard; Roy, Ranjan. Special functions. Encyclopedia of\n Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. xvi+664\n pp. ISBN: 0-521-62321-9; 0-521-78988-5
  • Biane, Philippe; Pitman, Jim; Yor, Marc. Probability laws related to the Jacobi theta\n and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. (N.S.) 38\n (2001), no. 4, 435–465 (electronic).
  • Chaumont, L.; Yor, M. Exercises in probability. A guided tour from measure theory to\n random processes, via conditioning. Cambridge Series in Statistical and Probabilistic\n Mathematics, 13. Cambridge University Press, Cambridge, 2003. xvi+236 pp. ISBN:\n 0-521-82585-7
  • Lamperti, John. An occupation time theorem for a class of stochastic processes. Trans.\n Amer. Math. Soc. 88 1958 380–387.
  • Lebedev, N. N. Special functions and their applications. Revised edition, translated\n from the Russian and edited by Richard A. Silverman. Unabridged and corrected republication.\n Dover Publications, Inc., New York, 1972. xii+308 pp.
  • Lévy, Paul. OE uvres de Paul Lévy. Vol. VI. (French) [Works of Paul Levy. Vol.\n VI] Théorie des jeux. [Game theory] Published under the direction of Daniel Dugué with\n the collaboration of Paul Deheuvels and Michel Ibéro. Gauthier-Villars, Paris, 1980. 423\n pp. (1 plate). ISBN: 2-04-010962-5
  • Pitman, Jim; Yor, Marc. Infinitely divisible laws associated with hyperbolic\n functions. Canad. J. Math. 55 (2003), no. 2, 292–330.
  • Pitman, Jim; Yor, Marc. Level crossings of a Cauchy process. Ann. Probab. 14 (1986),\n no. 3, 780–792.
  • J.P. Serre, Cours d'arithmétique, Collection SUP, P.U.F., Paris, 1970.
  • H. M. Srivasta, Junesang Choi, Series associated with the Zeta and Related Functions,\n 2006, Kluwer Academic Publishers, Dordrecht, 2001.