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Integrals over polytopes, multiple zeta values and polylogarithms, and Euler’s constant. (English) Zbl 1172.11028
The integral I n = 0 1 1-y 1 (-lnxy) n xydxdy is shown to be a sum of multiple zeta values, I n =n! k=0 n ζ(n-k+2,{1} k ), and I n =P(ζ(2),ζ(3),,ζ(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 ζ(n,{1} k ) as a polynomial in ζ(2),,ζ(n+k) with rational coefficients, and the interesting relation ζ(k+2,{1} l )=ζ(l+2,{1} k ). The asymptotic expansion I n n! k=1 2k k1 k n+2 is derived. A formula for I -1 is derived, and a similar formula for Euler’s constant γ. The authors discuss a number of examples.

MSC:
11M32Multiple Dirichlet series, etc.
33B30Higher logarithm functions