Kyoto Journal of Mathematics

Special values of the Riemann zeta function via arcsine random variables

Takahiko Fujita

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In this paper, using arcsine variables, we get a new elementary proof of ζ(2)=π2/6, known as the Basel problem, and the Euler formula. Using exponential variables, we get an Euler-like formula. We can also solve the Basel problem by using Wigner’s semicircle law and the Legendre generating function.

Article information

Kyoto J. Math., Volume 55, Number 3 (2015), 673-686.

Received: 12 October 2011
Revised: 14 August 2014
Accepted: 29 August 2014
First available in Project Euclid: 9 September 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 11M35: Hurwitz and Lerch zeta functions 60E05: Distributions: general theory

Basel problem Euler formula Legendre generating function Wigner’s semicircle law


Fujita, Takahiko. Special values of the Riemann zeta function via arcsine random variables. Kyoto J. Math. 55 (2015), no. 3, 673--686. doi:10.1215/21562261-3089145.

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