Challenging integral: Evaluate $\int_0^1\frac{\ln^3(1-x)\operatorname{Li}_3(x)}{x}dx$

Solution 1:

We can write $$ \ln(1-x)=-\int_0^x \frac{dt}{1-t},\hspace{7mm}\operatorname{Li}_3(x)=\int_{0\leq t_1\leq t_2\leq t_3\leq x}\frac{dt_1\,dt_2\,dt_3}{(1-t_1)t_2t_3}. $$ We can multiply out $\ln(1-x)^3\operatorname{Li}_3(x)$ and break the result into a sum over the different possible orderings of the variables of integration. This will allow us to write $I$ as an integer linear combination of terms of the form $$ \int_{0\leq t_1\leq t_2\leq t_3\leq t_4\leq t_5\leq t_6\leq t_7\leq 1}\frac{dt_1\,dt_2\,dt_3\,dt_4\,dt_5\,dt_6\,dt_7}{f_1(t_1)f_2(t_2)f_3(t_3)f_4(t_4)f_5(t_5)f_6(t_6)f_7(t_7)}, $$ where three of the functions $f_i$ are $f_i(x)=x$ and the other four are $f_i(x)=1-x$. Each of these latter integrals is a multiple zeta value of depth $4$ and weight $7$. It's a bit messy, but I find $$ I = -24\zeta(4, 1, 1, 1)-18\zeta(3, 2, 1, 1)-18\zeta(2, 3, 1, 1)-12\zeta(3, 1, 2, 1)-12\zeta(2, 2, 2, 1)-12\zeta(2, 1, 3, 1)-6\zeta(3, 1, 1, 2)-6\zeta(2, 2, 1, 2)-6\zeta(2, 1, 2, 2)-6\zeta(2, 1, 1, 3). $$ Every multiple zeta value of weight $7$ can be written in terms of the Riemann zeta function. There are a number of different ways to work out these expressions. One way is to use the generalized double-shuffle relations (see the paper [1]), and the expressions have been tabulated here. The result is $$ I=-\frac{327}{8}\zeta(7)+\frac{21}{5}\zeta(2)^{2}\zeta(3)+12\zeta(2)\zeta(5). $$

[1] Ihara, Kentaro; Kaneko, Masanobu; Zagier, Don, Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142, No. 2, 307-338 (2006). ZBL1186.11053.

Solution 2:

This post is a supplement to the nice answer from @JulianRosen. We go into some details thereby confirming his results, the intermediate result as well as the final one. As this post is rather lengthy albeit simple, here is short overview:

• Step 1: Representation of $I$ by multiple zeta values (MZV) confirming Julian Rosen's intermediate result.

• Step 2: Simplification of $I$ by application of a sum relation and the duality theorem for MZVs.

• Step 3: We make a shortcut and use EZ-Face, an MZV calculator, to find linear dependencies of MZVs. This way we obtain and thereby confirm Julian Rosen's final result.

Step1: Representation of $I$ by multiple zeta values (MZV)

We start with \begin{align*} I&=\int_0^1\frac{1}{x}\ln^3(1-x)\mathrm{Li}_3(x)\,dx\\ &=\int_0^1\frac{1}{x}\left(-\int_0^x\frac{dt}{1-t}\right)^3\int_{0<t_1<t_2<t_3<x}\frac{dt_1\,dt_2\,dt_3}{(1-t_1)t_2t_3}\,dx\\ &=-\int_{{0<t_1<t_2<t_3<t_7<1}\atop{0<t_4,t_5,t_6<t_7<1}}\frac{dt_1\,dt_2\,dt_3\,dt_4\,dt_5\,dt_6\,dt_7}{(1-t_1)t_2t_3(1-t_4)(1-t_5)(t-t_6)t_7}\tag{1} \end{align*}

In (1) we rewrite the integral to easily see how the region of integration \begin{align*} \{(t_1,t_2,t_3,t_4,t_5,t_6,t_7): 0<t_1<t_2<t_3<t_7<1,0<t_4,t_5,t_6<t_7<1\}\tag{2} \end{align*} is to split in order to obtain an integral representations of the MZVs of the form \begin{align*} \int_{0<t_1<t_2<t_3<t_4<t_5<t_6<t_7<1}\frac{dt_1\,dt_2\,dt_3\,dt_4\,dt_5\,dt_6\,dt_7}{(1-t_1)t_2t_3(1-t_4)(1-t_5)(t-t_6)t_7} \end{align*} See e.g. chapter 7 of Lectures on Multiple Zeta Values by W. Zudilin for more information.

Each of the following integrals has a factor $6$ since according to the region of integration in (2) we can permute $t_4,t_5,t_6$.

We obtain from (1) \begin{align*} I&=(-6)\left(\int_{0<\color{blue}{t_1}<t_2<t_3<\color{blue}{t_4}<\color{blue}{t_5}<\color{blue}{t_6}<t_7<1}+\int_{0<\color{blue}{t_1}<t_2<\color{blue}{t_4}<t_3<\color{blue}{t_5}<\color{blue}{t_6}<t_7<1}\right.\\ &\qquad\qquad+\int_{0<\color{blue}{t_1}<\color{blue}{t_4}<t_2<t_3<\color{blue}{t_5}<\color{blue}{t_6}<t_7<1}+\int_{0<\color{blue}{t_4}<\color{blue}{t_1}<t_2<t_3<\color{blue}{t_5}<\color{blue}{t_6}<t_7<1}\\ &\qquad\qquad+\int_{0<\color{blue}{t_1}<t_2<\color{blue}{t_4}<\color{blue}{t_5}<t_3<\color{blue}{t_6}<t_7<1}+\int_{0<\color{blue}{t_1}<\color{blue}{t_4}<t_2<\color{blue}{t_5}<t_3<\color{blue}{t_6}<t_7<1}\\ &\qquad\qquad+\int_{0<\color{blue}{t_4}<\color{blue}{t_1}<t_2<\color{blue}{t_5}<t_3<\color{blue}{t_6}<t_7<1}+\int_{0<\color{blue}{t_1}<\color{blue}{t_4}<\color{blue}{t_5}<t_2<t_3<\color{blue}{t_6}<t_7<1}\\ &\qquad\qquad+\int_{0<\color{blue}{t_4}<\color{blue}{t_1}<\color{blue}{t_5}<t_2<t_3<\color{blue}{t_6}<t_7<1}+\int_{0<\color{blue}{t_4}<\color{blue}{t_5}<\color{blue}{t_1}<t_2<t_3<\color{blue}{t_6}<t_7<1}\\ &\qquad\qquad+\int_{0<\color{blue}{t_1}<t_2<\color{blue}{t_4}<\color{blue}{t_5}<\color{blue}{t_6}<t_3<t_7<1}+\int_{0<\color{blue}{t_1}<\color{blue}{t_4}<t_2<\color{blue}{t_5}<\color{blue}{t_6}<t_3<t_7<1}\\ &\qquad\qquad+\int_{0<\color{blue}{t_4}<\color{blue}{t_1}<t_2<\color{blue}{t_5}<\color{blue}{t_6}<t_3<t_7<1}+\int_{0<\color{blue}{t_1}<\color{blue}{t_4}<\color{blue}{t_5}<t_2<\color{blue}{t_6}<t_3<t_7<1}\\ &\qquad\qquad+\int_{0<\color{blue}{t_4}<\color{blue}{t_1}<\color{blue}{t_5}<t_2<\color{blue}{t_6}<t_3<t_7<1}+\int_{0<\color{blue}{t_4}<\color{blue}{t_5}<\color{blue}{t_1}<t_2<\color{blue}{t_6}<t_3<t_7<1}\\ &\qquad\qquad+\int_{0<\color{blue}{t_1}<\color{blue}{t_4}<\color{blue}{t_5}<\color{blue}{t_6}<t_2<t_3<t_7<1}+\int_{0<\color{blue}{t_4}<\color{blue}{t_1}<\color{blue}{t_5}<\color{blue}{t_6}<t_2<t_3<t_7<1}\\ &\qquad\qquad\left.+\int_{0<\color{blue}{t_4}<\color{blue}{t_5}<\color{blue}{t_1}<\color{blue}{t_6}<t_2<t_3<t_7<1}+\int_{0<\color{blue}{t_4}<\color{blue}{t_5}<\color{blue}{t_6}<\color{blue}{t_1}<t_2<t_3<t_7<1}\right)\tag{3}\\ &\qquad\qquad\qquad\frac{dt_1\,dt_2\,dt_3\,dt_4\,dt_5\,dt_6\,dt_7}{(1-t_1)t_2t_3(1-t_4)(1-t_5)(t-t_6)t_7}\\ &=(-6)\left(4\int_{0<\color{blue}{t_1}<\color{blue}{t_4}<\color{blue}{t_5}<\color{blue}{t_6}<t_2<t_3<t_7<1}\frac{dt_1\,dt_4\,dt_5\,dt_6\,dt_2\,dt_3\,dt_7}{(1-t_1)(1-t_4)(1-t_5)(t-t_6)t_2t_3t_7}\right.\\ &\qquad\qquad+3\int_{0<\color{blue}{t_1}<\color{blue}{t_4}<\color{blue}{t_5}<t_2<\color{blue}{t_6}<t_3<t_7<1}\frac{dt_1\,dt_4\,dt_5\,dt_2\,dt_6\,dt_3\,dt_7}{(1-t_1)(1-t_4)(1-t_5)t_2(t-t_6)t_3t_7}\\ &\qquad\qquad+3\int_{0<\color{blue}{t_1}<\color{blue}{t_4}<\color{blue}{t_5}<t_2<t_3<\color{blue}{t_6}<t_7<1}\frac{dt_1\,dt_4\,dt_5\,dt_2\,dt_3\,dt_6\,dt_7}{(1-t_1)(1-t_4)(1-t_5)t_2t_3(t-t_6)t_7}\\ &\qquad\qquad+2\int_{0<\color{blue}{t_1}<\color{blue}{t_4}<t_2<\color{blue}{t_5}<\color{blue}{t_6}<t_3<t_7<1}\frac{dt_1\,dt_4\,dt_2\,dt_5\,dt_6\,dt_3\,dt_7}{(1-t_1)(1-t_4)t_2(1-t_5)(t-t_6)t_3t_7}\\ &\qquad\qquad+2\int_{0<\color{blue}{t_1}<\color{blue}{t_4}<t_2<\color{blue}{t_5}<t_3<\color{blue}{t_6}<t_7<1}\frac{dt_1\,dt_4\,dt_2\,dt_5\,dt_3\,dt_6\,dt_7}{(1-t_1)(1-t_4)t_2(1-t_5)t_3(t-t_6)t_7}\\ &\qquad\qquad+2\int_{0<\color{blue}{t_1}<\color{blue}{t_4}<t_2<t_3<\color{blue}{t_5}<\color{blue}{t_6}<t_7<1}\frac{dt_1\,dt_4\,dt_2\,dt_3\,dt_5\,dt_6\,dt_7}{(1-t_1)(1-t_4)t_2t_3(1-t_5)(t-t_6)t_7}\\ &\qquad\qquad+\int_{0<\color{blue}{t_1}<t_2<\color{blue}{t_4}<\color{blue}{t_5}<\color{blue}{t_6}<t_3<t_7<1}\frac{dt_1\,dt_2\,dt_4\,dt_5\,dt_6\,dt_3\,dt_7}{(1-t_1)t-2(1-t_4)(1-t_5)(t-t_6)t_3t_7}\\ &\qquad\qquad+\int_{0<\color{blue}{t_1}<t_2<\color{blue}{t_4}<\color{blue}{t_5}<t_3<\color{blue}{t_6}<t_7<1}\frac{dt_1\,dt_2\,dt_4\,dt_5\,dt_3\,dt_6\,dt_7}{(1-t_1)t_2(1-t_4)(1-t_5)t_3(t-t_6)t_7}\\ &\qquad\qquad+\int_{0<\color{blue}{t_1}<t_2<\color{blue}{t_4}<t_3<\color{blue}{t_5}<\color{blue}{t_6}<t_7<1}\frac{dt_1\,dt_2\,dt_4\,dt_3\,dt_5\,dt_6\,dt_7}{(1-t_1)t_2(1-t_4)t_3(1-t_5)(t-t_6)t_7}\\ &\qquad\qquad\left.+\int_{0<\color{blue}{t_1}<t_2<t_3<\color{blue}{t_4}<\color{blue}{t_5}<\color{blue}{t_6}<t_7<1}\frac{dt_1\,dt_2\,dt_3\,dt_4\,dt_5\,dt_6\,dt_7}{(1-t_1)t_2t_3(1-t_4)(1-t_5)(t-t_6)t_7}\right)\tag{4}\\ &=(-6)\left(4\zeta(4,1,1,1)+3\zeta(3,2,1,1)+3\zeta(2,3,1,1)\right.\\ &\qquad\qquad+2\zeta(3,1,2,1)+2\zeta(2,2,2,1)+2\zeta(2,1,3,1)\\ &\qquad\qquad\left.+\zeta(3,1,1,2)+\zeta(2,2,1,2)+\zeta(2,1,2,2)+\zeta(2,1,1,3)\right)\tag{5} \end{align*} in accordance with the intermediate result of Julian Rosen.

In (3) we marked the indices $t_1,t_4,t_5,t_6$ corresponding to $\frac{1}{1-t_j}$ blue, to better see the structure of the integral. In (4) we collect all terms with the same structure (appropriately substituing indices thereby).

Step 2: Simplification of $I$

Here we recall the Sum theorem (Theorem 2.5 in W. Zudilins paper): For any integer $s>1$ and $l\geq 1$ the following holds, \begin{align*} \sum_{{s_1>1,s_2\geq 1,\ldots,s_l\geq 1}\atop{s_1+s_2+\cdots+s_l=s}}\zeta(s_1,s_2,\ldots,s_l)=\zeta(s) \end{align*}

Setting $l=4$ and $s=7$ we obtain \begin{align*} \zeta(7)&=\zeta(4,1,1,1)+\zeta(3,2,1,1)+\zeta(2,3,1,1)\\ &+\zeta(3,1,2,1)+\zeta(2,2,2,1)+\zeta(2,1,3,1)\\ &+\zeta(3,1,1,2)+\zeta(2,2,1,2)+\zeta(2,1,2,2)\\ &+\zeta(2,1,1,3) \end{align*}

Putting this relation into (5) we obtain

\begin{align*} \color{blue}{I}&\color{blue}{=(-6)\left(3\zeta(4,1,1,1)+2\zeta(3,2,1,1)+2\zeta(2,3,1,1)\right.}\\ &\qquad\qquad\quad\color{blue}{+\zeta(3,1,2,1)+\zeta(2,2,2,1)+\zeta(2,1,3,1)}\\ &\qquad\qquad\quad\color{blue}{\left.+\zeta(7)\right)}\tag{6} \end{align*}

Next we apply the duality theorem (Theorem 3.7 in W. Zudilins paper) and replace this way the MZVs with length $4$ with MZVs of length $3$.

We obtain from (6) \begin{align*} \zeta(4,1,1,1)&=Z(x^3y\cdot y\cdot y\cdot y)=Z(x^3y^4)=Z(x^4y^3)=\zeta(5,1,1)\\ \zeta(3,2,1,1)&=Z(x^2y\cdot xy\cdot y\cdot y)=Z(x^2yxy^3)=Z(x^3yxy^2)=\zeta(4,2,1)\\ \zeta(2,3,1,1)&=Z(xy\cdot x^2y\cdot y\cdot y)=Z(xyx^2y^3)=Z(x^3y^2xy)=\zeta(4,1,2)\\ \zeta(3,1,2,1)&=Z(x^2y\cdot y\cdot xy\cdot y)=Z(x^2y^2xy^2)=Z(x^2yx^2y^2)=\zeta(3,3,1)\\ \zeta(2,2,2,1)&=Z(xy\cdot xy\cdot xy\cdot y)=Z(xyxyxy^2)=Z(x^2yxyxy)=\zeta(3,2,2)\\ \zeta(2,1,3,1)&=Z(xy\cdot y\cdot x^2y\cdot y)=Z(xy^2x^2y^2)=Z(x^2y^2x^2y)=\zeta(3,1,3)\tag{7} \end{align*}

From (6) and (7) we derive a simpler representation \begin{align*} \color{blue}{I}&\color{blue}{=(-6)\left(3\zeta(5,1,1)+2\zeta(4,2,1)+2\zeta(4,1,2)\right.}\\ &\qquad\color{blue}{+\zeta(3,3,1)+\zeta(3,2,2)+\zeta(3,1,3)}\\ &\qquad\color{blue}{\left.+\zeta(7)\right)}\tag{8} \end{align*}

Step 3: Representation of $I$ with single zeta values

We know that MZVs of weight $7$ are in the $\mathbb{Q}$-linear span of $\zeta(7),\zeta(2)^2\zeta(3)$ and $\zeta(2)\zeta(5)$. See e.g. exercise 3.3 (iv) in W. Zudilins paper. We use EZ-Face to find the linear relations.

E.g. typing the input

\begin{align*} \mathrm{lindep}([z(7),z(2)*z(2)*z(3), z(2)*z(5), z(5,1,1)]) \end{align*}

The output is:

\begin{align*} [10, -1., -4.,-2.] \end{align*}

which means \begin{align*} 10\zeta(7)-\zeta(2)^2\zeta(3)-4\zeta(2)\zeta(5)-2\zeta(5,1,1,1)=0 \end{align*}

We find this way the coefficients of the linear expressions for all MZVs of length $4$ in (8) as

$$ \begin{array}{rrr|r|c} \zeta(7)&\zeta(2)^2\zeta(3)&\zeta(2)\zeta(5)&\zeta(\mathrm{arg})&\mathrm{arg}\\ \hline 10&-1&-4&-2&(5,1,1)\\ 1105&-112&-440&80&(4,2,1)\\ -5&12&-20&8&(4,1,2)\\ -61&0&36&8&(3,3,1)\\ -785&-72&600&80&(3,2,2)\\ -5&2&0&-20&(3,1,3) \end{array} $$

Substituting the MZVs in (8) with the linear representation of the table we finally obtain \begin{align*} \color{blue}{I}&\color{blue}{=(-6)\left(3\zeta(5,1,1)+2\zeta(4,2,1)+2\zeta(4,1,2)\right.}\\ &\qquad\color{blue}{+\zeta(3,3,1)+\zeta(3,2,2)+\zeta(3,1,3)}\\ &\qquad\color{blue}{\left.+\zeta(7)\right)}\\ &=(-6)\left(\zeta(7)\left(3\cdot\frac{10}{2}-2\cdot\frac{1105}{80}+2\cdot\frac{5}{8}+\frac{61}{8}+\frac{785}{80}-\frac{5}{20}+1\right)\right.\\ &\qquad\qquad\quad+\zeta(2)^2\zeta(3)\left(-3\cdot\frac{1}{2}+2\cdot\frac{112}{80}-2\cdot\frac{12}{8}+0+\frac{72}{80}+\frac{2}{20}\right)\\ &\qquad\qquad\left.\quad+\zeta(2)\zeta(5)\left(-3\cdot\frac{4}{2}+2\cdot\frac{440}{80}+2\cdot\frac{20}{8}-\frac{36}{8}-\frac{600}{80}+0\right)\right)\\ &\color{blue}{=-\frac{327}{8}\zeta(7)+\frac{21}{5}\zeta(2)^2\zeta(3)+12\zeta(2)\zeta(5)}\\ \end{align*}

in accordance with Julian Rosen's result.