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Integrals over polytopes, multiple zeta values and polylogarithms, and Euler’s constant. (English. Russian original) Zbl 1172.11028
The integral $I_n=\int_0^1\int_{1-y}^1\frac{(-\ln xy)^n}{xy}\,dx\,dy$ is shown to be a sum of multiple zeta values, $I_n=n!\sum_{k=0}^n\zeta(n-k+2,\{1\}_k)$, and $I_n=P(\zeta(2),\zeta(3),\dots,\zeta(n+2))$ for some polynomial with rational coefficients, which is explicitly given in the restatement of Theorem 1 on page 573. Combining this, it allows an expression for $\zeta(n,\{1\}_k)$ as a polynomial in $\zeta(2),\dots,\zeta(n+k)$ with rational coefficients, and the interesting relation $\zeta(k+2,\{1\}_l)=\zeta(l+2,\{1\}_k)$. The asymptotic expansion $\frac{I_n}{n!}\sim\sum_{k=1}^\infty\binom{2k}{k}\frac1{k^{n+2}}$ is derived. A formula for $I_{-1}$ is derived, and a similar formula for Euler’s constant $\gamma$. The authors discuss a number of examples.

11M32Multiple Dirichlet series, etc.
33B30Higher logarithm functions
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