Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant

Details Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant

2007-05-05

arxiv.org/abs/0705.0732v2

Math. Notes (in English) 84 (2008) pp. 568-583; Mat. Zametki (in Russian) 84:4 (2008) pp. 609-626

Mathematics

Number Theory

19 pages, to appear in Mat Zametki. Ver 2.: Added Remark 3 on a Chen (Drinfeld-Kontsevich) iterated integral; simplified Propo

Scientific paper

10.1134/S0001434608090290

Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler's constant. In the final section, we evaluate more general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral -- over some polytopes that are higher-dimensional analogs of $T$. This leads to a relation between certain multiple polylogarithm values and multiple zeta values.

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