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Question Number 155136 by amin96 last updated on 25/Sep/21

$$\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{t}\mathit{h}\mathit{a}\mathit{t}$$$$\frac{{\mathbf{log}}^2\left(2\right)}{2}\underset{\mathit{n}=1}{\sum ^{\mathrm{\infty }}}\frac{\mathit{\phi }\left(\mathit{n}\right)}{{(\mathit{n}+1)}^2{2}^{\mathit{n}}}+\mathbf{log}\left(2\right)\underset{\mathit{n}=1}{\sum ^{\mathrm{\infty }}}\frac{\mathit{\phi }\left(n\right)}{{(\mathit{n}+1)}^3{2}^{\mathit{n}}}+\underset{\mathit{n}=1}{\sum ^{\mathrm{\infty }}}\frac{\mathit{\phi }\left(n\right)}{{(\mathit{n}+1)}^4{2}^{\mathit{n}}}=$$$$=\frac{23}{8}\mathit{\zeta }\left(6\right)-2{\mathit{\zeta }}^2\left(3\right)-\frac{1}{18}{\mathbf{log}}^6\left(2\right)$$$$\mathit{m}.\mathit{A}$$

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