# 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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