Probabilistic proofs of Euler identities. (English) Zbl 1295.33002
Probability techniques are used to prove three well-known formulae: a formula relating the Riemann-zeta function to the tangent numbers; a formula relating the function
\[
L_{\chi_4}(2n+1)=\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^{2n+1}}
\]
to the Euler numbers; and Euler’s infinite product formula for the sine function. The hyperbolic secant distribution is used to derive the formulas.
Reviewer: Jeremy Wade (Pittsburg)
MSC:
| 33B10 | Exponential and trigonometric functions |
| 01A50 | History of mathematics in the 18th century |
| 60F05 | Central limit and other weak theorems |
| 60E05 | Probability distributions: general theory |
Keywords:
Basel problem; hyperbolic secant distribution; Euler number; tangent number; Euler’s sine productReferences:
| [1] | Baten, W. D. (1934). The probability law for the sum of \(n\) independent variables, each subject to the law \((1/(2h)) \text{sech}( \pi x/(2h))\). Bull. Amer. Math. Soc. 40, 284-290. · Zbl 0009.21903 · doi:10.1090/S0002-9904-1934-05852-X |
| [2] | Bourgade, P., Fujita, T. and Yor, M. (2007). Euler’s formulae for \(\zeta(2n)\) and products of Cauchy variables. Electron. Commun. Prob. 12, 73-80. · Zbl 1129.60088 · doi:10.1214/ECP.v12-1244 |
| [3] | Bradley, R. E., D’Antonio, L. A. and Sandifer, C. E. (eds) (2007). Euler at 300. An Appreciation . Mathematical Association of America, Washington, DC. · Zbl 1165.01007 |
| [4] | Chapman, R. (2003). Evaluating \(\zeta(2)\). Preprint. Available at http://www.uam.es/personal_pdi/ciencias/cillerue/Curso/zeta2.pdf Dunham, W. (1999). Euler: The Master of Us All . Mathematical Association of America, Washington, DC. |
| [5] | Feller, W. (1966). An Introduction to Probability Theory and Its Applications , Vol. 2. John Wiley, New York. · Zbl 0138.10207 |
| [6] | Gordon, L. (1994). A stochastic approach to the gamma function. Amer. Math. Monthly 101, 858-865. · Zbl 0823.33001 · doi:10.2307/2975134 |
| [7] | Harkness, W. L. and Harkness, M. L. (1968). Generalized hyperbolic secant distributions. J. Amer. Statist. Assoc. 63, 329-337. · Zbl 0155.27204 · doi:10.2307/2283852 |
| [8] | Harper, J. D. (2003). Another simple proof of \(1+1/2^2+1/3^2+\dots = \pi^2/6\). Amer. Math. Monthly 110, 540-541. · Zbl 1043.40500 · doi:10.2307/3647912 |
| [9] | Havil, J. (2003). Gamma: Exploring Euler’s Constant . Princeton University Press. · Zbl 1023.11001 |
| [10] | Hofbauer, J. (2002). A simple proof of \(1+1/2^2+1/3^2+\dots = \pi^2/6\) and related identities. Amer. Math. Monthly 109, 196-200. · Zbl 1022.40002 · doi:10.2307/2695334 |
| [11] | Holst, L. (2012). A proof of Euler’s infinite product for the sine. Amer. Math. Monthly 119, 518-521. · Zbl 1264.33001 · doi:10.4169/amer.math.monthly.119.06.518 |
| [12] | Kalman, D. (1993). Six ways to sum a series. College Math. J. 24, 402-421. · Zbl 1266.40006 |
| [13] | 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. · Zbl 1245.33006 · doi:10.4169/amer.math.monthly.119.01.042 |
| [14] | Marshall, T. (2010). A short proof of \(\zeta(2)=\pi^2/6\). Amer. Math. Monthly 117, 352-353. · Zbl 1275.11119 · doi:10.4169/000298910X480810 |
| [15] | Pace, L. (2011). Probabilistically proving that \(\zeta(2)=\pi^2/6\). Amer. Math. Monthly 118, 641-643. · Zbl 1227.00039 · doi:10.4169/amer.math.monthly.118.07.641 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
