Kyoto Journal of Mathematics

Special values of the Riemann zeta function via arcsine random variables

Takahiko Fujita

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1441824044

Digital Object Identifier
doi:10.1215/21562261-3089145

Mathematical Reviews number (MathSciNet)
MR3395986

Zentralblatt MATH identifier
1326.11045

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

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

Citation

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. https://projecteuclid.org/euclid.kjm/1441824044


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References

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