Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T03:57:17.710Z Has data issue: false hasContentIssue false
  • English
  • Français

Probabilistic Proofs of Euler Identities

Part of: Elementary classical functions History of mathematics and mathematicians

Published online by Cambridge University Press:  30 January 2018

Lars Holst
Lars Holst*
Affiliation:
Royal Institute of Technology
*
∗ Postal address: Department of Mathematics, Royal Institute of Technology, SE-10044 Stockholm, Sweden. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Formulae for ζ(2n) and Lχ4(2n + 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π2 / 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.

Keywords

Basel problemhyperbolic secant distributionEuler numbertangent numberEuler's sine product

MSC classification

Primary: 33B10: Exponential and trigonometric functions
Secondary: 01A50: 18th century
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1206 - 1212
Copyright
© Applied Probability Trust 

References

Baten, W. D. (1934). The probability law for the sum of n independent variables, each subject to the law (1/(2h)) sech(π x/(2h)). Bull. Amer. Math. Soc. 40, 284–290.Google Scholar
Bourgade, P., Fujita, T. and Yor, M. (2007). Euler's formulae for ζ(2n) and products of Cauchy variables. Electron. Commun. Prob. 12, 73–80.Google Scholar
Bradley, R. E., D'Antonio, L. A. and Sandifer, C. E. (eds) (2007). Euler at 300. An Appreciation. Mathematical Association of America, Washington, DC.CrossRefGoogle Scholar
Chapman, R. (2003). Evaluating ζ(2). Preprint. Available at http://www.uam.es/personal_pdi/ciencias/cillerue/Curso/zeta2.pdf Google Scholar
Dunham, W. (1999). Euler: The Master of Us All. Mathematical Association of America, Washington, DC.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. 2. John Wiley, New York.Google Scholar
Gordon, L. (1994). A stochastic approach to the gamma function. Amer. Math. Monthly 101, 858–865.Google Scholar
Harkness, W. L. and Harkness, M. L. (1968). Generalized hyperbolic secant distributions. J. Amer. Statist. Assoc. 63, 329–337.Google Scholar
Harper, J. D. (2003). Another simple proof of 1+1/22+1/32+… = π2/6. Amer. Math. Monthly 110, 540–541.Google Scholar
Havil, J. (2003). Gamma: Exploring Euler's Constant. Princeton University Press.Google Scholar
Hofbauer, J. (2002). A simple proof of 1+1/22+1/32+… = π2/6 and related identities. Amer. Math. Monthly 109, 196–200.Google Scholar
Holst, L. (2012). A proof of Euler's infinite product for the sine. Amer. Math. Monthly 119, 518–521.Google Scholar
Kalman, D. (1993). Six ways to sum a series. College Math. J. 24, 402–421.CrossRefGoogle Scholar
Kalman, D. and McKinzie, M. (2012). Another way to sum a series: generating functions, Euler, and the dilog function. Amer. Math. Monthly 119, 42–51.Google Scholar
Marshall, T. (2010). A short proof of ζ(2)=π2/6 . Amer. Math. Monthly 117, 352–353.Google Scholar
Pace, L. (2011). Probabilistically proving that ζ(2)=π2/6 . Amer. Math. Monthly 118, 641–643.CrossRefGoogle Scholar