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Some Results Involving Euler-Type Integrals and Dilogarithm Values

  • Lubomir Markov
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 728)

Abstract

The claim that \( \text {Li}_2\Big (- \displaystyle \frac{\sqrt{5}+1}{2}\Big ) = -\displaystyle \frac{\,\, \pi ^2}{10} + \displaystyle \frac{1}{2} \log ^2 \Big ( \frac{\sqrt{5}+1}{2}\Big ),\) circulating in the dilogarithm literature since at least 1958, is wrong. We derive the correct value \(\,\text {Li}_2\bigg ( -\displaystyle \frac{\sqrt{5}+1}{2}\bigg ) = -\displaystyle \frac{\,\, \pi ^2}{10} - \log ^2 \bigg ( \displaystyle \frac{\sqrt{5}+1}{2} \bigg )\,\) and use it to obtain several formulas for \(\pi ^2\) in terms of dilogarithm values at the “golden relatives” \(\, \displaystyle \frac{1}{\phi ^2}, \, \displaystyle \frac{1}{\phi }, \, -\,\displaystyle \frac{1}{\phi }, -\,\phi \ \) of \(\,\phi \,= \,\displaystyle \frac{\sqrt{5}+1}{2}.\) We also sum the series \(\,\displaystyle \sum _{n=0}^\infty \displaystyle \frac{G_N(n)}{(2n+1)^3} \,\) and \(\,\displaystyle \sum _{n=1}^\infty \frac{H_N(n)}{n^3} \,\) in terms of Euler-type integrals \(\,\displaystyle \int _0^{\frac{\pi }{2}} x^{M} \log (\sin x)\, \mathrm{d}x, \,\) where \(\,G_N(n)\,\) and \(\,H_N(n)\,\) are the quantities appearing in the Borwein-Chamberland expansions of \( \,\arcsin ^{2N+1}(z) \,\) and \(\, \arcsin ^{2N}(z), \,\) respectively. As special cases we obtain very simple proofs of Euler’s equation
$$\,\zeta (3) = \frac{2\pi ^2 }{7} \log 2 + \frac{16}{7} \int _0^{\frac{\pi }{2}} x \log (\sin x) {\text {d}} x \,$$
and of the similar formula
$$\,\zeta (3) = \frac{2\pi ^2 }{9} \log 2 + \frac{16}{3\pi } \int _0^{\frac{\pi }{2}} x^2 \log (\sin x) {\text {d}}x.$$

Keywords

Dilogarithm Golden ratio Riemann zeta function Euler-type integrals Borwein-Chamberland expansions Euler’s equation for \(\zeta (3)\) Wallis integrals 

Notes

Acknowledgements

The author expresses his gratitude to the referee who pointed out to him the fact that the value for \(\text {Li}_2\big ( -\phi \big )\) is given on the Internet.

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and CSBarry UniversityMiami ShoresUSA

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