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Question Number 123387 by mnjuly1970 last updated on 25/Nov/20

$$...nicecalculus...$$$$provethat::$$$$\mathrm{\Omega }={\int }_0^1\frac{{\left(ln\left(x\right)\right)}^2{li}_3\left(x\right)}{1-x}dx$$$$\stackrel{???}{=}{\zeta }^2\left(3\right)-\zeta \left(6\right)✓$$

Commented by talminator2856791 last updated on 25/Nov/20

$$\mathrm{what}\mathrm{is}{\mathrm{li}}_3?$$

Commented by mnjuly1970 last updated on 25/Nov/20

$${li}_3\left(x\right)={\int }_0^x\frac{{li}_2\left(t\right)}{t}dt=\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{x^n}{n^3}$$$$trilogarithmfunction.$$$${li}_z\left(x\right)=\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{x^n}{n^z}$$$$example:{li}_2\left(1\right)=\zeta \left(2\right)=\frac{{\pi }^2}{6}$$$${li}_3\left(1\right)=\zeta \left(3\right),{li}_2(-1)=-\eta \left(2\right)=-\frac{{\pi }^2}{12}$$$$,....$$

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