Some Results Involving Euler-Type Integrals and Dilogarithm Values

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

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