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Mat. Zametki, 2008, Volume 84, Issue 4, Pages 609–626 (Mi mz6140)  

This article is cited in 2 scientific papers (total in 2 papers)

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

J. Sondow, S. A. Zlobina

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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 linear combinations of multiple zeta values, and as polynomials 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, and study integrals, one of which is the iterated Chen (Drinfeld–Kontsevich) integral, over some polytopes that are higher-dimensional analogs of $T$. The latter leads to a relation between certain multiple polylogarithm values and multiple zeta values.

Keywords: polytope, multiple zeta value, Riemann's zeta function, algebraic independence of numbers, polylogarithm, Abel summability, gamma function, meromorphic function

DOI: https://doi.org/10.4213/mzm6140

Full text: PDF file (554 kB)
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English version:
Mathematical Notes, 2008, 84:4, 568–583

Bibliographic databases:

UDC: 517.51
Received: 09.02.2007

Citation: J. Sondow, S. A. Zlobin, “Integrals over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant”, Mat. Zametki, 84:4 (2008), 609–626; Math. Notes, 84:4 (2008), 568–583

Citation in format AMSBIB
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\by J.~Sondow, S.~A.~Zlobin
\paper Integrals over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant
\jour Mat. Zametki
\yr 2008
\vol 84
\issue 4
\pages 609--626
\mathnet{http://mi.mathnet.ru/mz6140}
\crossref{https://doi.org/10.4213/mzm6140}
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\zmath{https://zbmath.org/?q=an:1172.11028}
\transl
\jour Math. Notes
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\vol 84
\issue 4
\pages 568--583
\crossref{https://doi.org/10.1134/S0001434608090290}
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  • https://doi.org/10.4213/mzm6140
  • http://mi.mathnet.ru/eng/mz/v84/i4/p609

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. A. Zlobin, “Special values of generalized polylogarithms”, J. Math. Sci., 182:4 (2012), 484–504  mathnet  crossref  mathscinet
    2. Lagarias J.C., “Euler's Constant: Euler's Work and Modern Developments”, Bull. Amer. Math. Soc., 50:4 (2013), 527–628  crossref  mathscinet  zmath  isi  elib  scopus
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