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

     ...  nice calculus...      prove  that ::          Ω=∫_0 ^( 1) (((ln(x))^2 li_3 (x))/(1−x)) dx                   =^(???) ζ^2 (3)−ζ(6) ✓

$$\:\:\:\:\:...\:\:{nice}\:{calculus}... \\ $$$$\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\left({ln}\left({x}\right)\right)^{\mathrm{2}} {li}_{\mathrm{3}} \left({x}\right)}{\mathrm{1}−{x}}\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{???} {=}\zeta^{\mathrm{2}} \left(\mathrm{3}\right)−\zeta\left(\mathrm{6}\right)\:\checkmark \\ $$

Commented by talminator2856791 last updated on 25/Nov/20

 what is li_3 ?

$$\:\mathrm{what}\:\mathrm{is}\:\mathrm{li}_{\mathrm{3}} ? \\ $$

Commented by mnjuly1970 last updated on 25/Nov/20

 li_3 (x) =∫_0 ^( x) ((li_2 (t))/t)dt=Σ_(n=1) ^∞ (x^n /n^3 )     trilogarithm function.     li_z (x)=Σ_(n=1) ^∞ (x^n /n^z )  example: li_2 (1)=ζ(2)=(π^2 /6)   li_3 (1)=ζ(3)  , li_2 (−1)=−η(2)=−(π^2 /(12))   ,....

$$\:{li}_{\mathrm{3}} \left({x}\right)\:=\int_{\mathrm{0}} ^{\:{x}} \frac{{li}_{\mathrm{2}} \left({t}\right)}{{t}}{dt}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} }{{n}^{\mathrm{3}} } \\ $$$$\:\:\:{trilogarithm}\:{function}. \\ $$$$\:\:\:{li}_{{z}} \left({x}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} }{{n}^{{z}} } \\ $$$${example}:\:{li}_{\mathrm{2}} \left(\mathrm{1}\right)=\zeta\left(\mathrm{2}\right)=\frac{\pi^{\mathrm{2}} }{\mathrm{6}} \\ $$$$\:{li}_{\mathrm{3}} \left(\mathrm{1}\right)=\zeta\left(\mathrm{3}\right)\:\:,\:{li}_{\mathrm{2}} \left(−\mathrm{1}\right)=−\eta\left(\mathrm{2}\right)=−\frac{\pi^{\mathrm{2}} }{\mathrm{12}}\: \\ $$$$,.... \\ $$

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