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Integrals over polytopes, multiple zeta values and polylogarithms, and Euler’s constant. (English. Russian original) Zbl 1172.11028

Math. Notes 84, No. 4, 568-583 (2008); translation from Mat. Zametki 84, No. 4, 609-626 (2008); erratum Math. Notes 84, No. 6, 887 (2008).
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.

MSC:

11M32 Multiple Dirichlet series and zeta functions and multizeta values
33B30 Higher logarithm functions
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References:

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