Some Results Involving Euler-Type Integrals and Dilogarithm Values

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


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
$$\,\zeta (3) = \frac{2\pi ^2 }{9} \log 2 + \frac{16}{3\pi } \int _0^{\frac{\pi }{2}} x^2 \log (\sin x) {\text {d}}x.$$


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



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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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and CSBarry UniversityMiami ShoresUSA

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