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Probabilistic Proofs of Euler Identities

Published online by Cambridge University Press:  30 January 2018

Lars Holst
Affiliation:
Royal Institute of Technology
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Abstract

Formulae for ζ(2n) and L χ4 (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.

Type
Research Article
Copyright
© Applied Probability Trust 

References

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