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research-article

A few conjectures about the multiple zeta values

Published: 26 July 2018 Publication History

Abstract

The multiple zeta values (MZVs) form a set of real numbers with, what we believe and module Zagier's Conjecture, a beautiful structure as an algebra over the rational numbers. In this paper we show partial solutions to five well known conjectures about the MZV obtained by using linear systems and shuffle products. We use computer algebra systems to study these conjectures and make our code available online.
We describe how to compute the shuffle minus stuffle product to obtain relations of the MZVs, together with the xtaylor algorithm, to verify the conjectures mentioned above. We note that it is only necessary to solve a linear system, not in MZV but in its powers and products, to find all the relations. We also made all the code available online.

References

[1]
F. Brown, On the decomposition of motivic multiple zeta values., arXiv:1102.1310.
[2]
F. Brown, Mixed Tate motives over Z., Annals of Mathematics 175 (2012), 949976.
[3]
P. Deligne, A. B. Goncharov, Groupes fondamentaux motiviques de Tate mixte., Ann. Sci. Ecole Norm.Sup. (4) 38 (2005), no. 1, 1--56.
[4]
M. Hoffman, The algebra of multiple harmonic series, J. Algebra 194:2 (1997), 477495.
[5]
M. Hoffman, Multiple Zeta Values and Euler Sums, http://www.usna.edu/Users/math/meh/biblio.html.
[6]
Ihara, K., Kaneko, M., Zagier, D., Derivation and double shuffle relations for multiple zeta values, Compositio Mathematica, 142(2), 307--338 (2006).
[7]
M. Kaneko, M. Noro, AND K. Tsurumaki, On a conjecture for the dimension of the space of the multiple zeta values, Software for Algebraic Geometry, IMA 148 (2008), 4758.
[8]
N. Nielsen, Handbuch der theorie der gammafunktion Published 1906 by Teubner in Leipzig .
[9]
H.M. Minh, Polylogarithms & evaluation transform, IMACS Symposium, Lille, June 1993.
[10]
H.M. Minh, G. Jacob, M. Petitot, and N. E. Oussous, Aspects combinatoires des polylogarithmes et des sommes dEulerZagier, S m. Lothar. Combin. 43 (1999)
[11]
H.N. Minh, M. Petitot Lyndon words, polylogarithms and the Riemann ζ function, Discrete Mathematics, Volume 217, Number 1, 28 April 2000, pp. 273--292(20).
[12]
H.N. Minh, M. Petitot, J. Van Der Hoeven, Shuffle algebra and polylogarithms, Proc. of FPSAC98, 10th International Conference on Formal Power Series and Algebraic Combinatorics,Toronto, June 1998.
[13]
D. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Algebra 58 (1979) 432--454.
[14]
R. Ree, Lie elements and an algebra associated with shuffles, Ann. Math. 68 (1958) 210220.
[15]
C, Reutenauer, Free Lie Algebras, London Mathematical Society Monographs, Clarendon Press (June 10, 1993).
[16]
M. Waldschmidt, Valeurs zeta multiples: une introduction, J. Th or. Nombres Bordeaux 12:2 (2000), 581595.
[17]
W. Zudilin, Algebraic relations for multiple zeta values, Russian Mathematical Surveys Volume 58, Number 1, 2001.
[18]
W. Zudilin, Arithmetic of linear forms involving odd zeta values, Journal Th. Nombres Bordeaux16 (2004), 251--291.

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Information & Contributors

Information

Published In

cover image ACM Communications in Computer Algebra
ACM Communications in Computer Algebra  Volume 52, Issue 1
March 2018
20 pages
ISSN:1932-2232
EISSN:1932-2240
DOI:10.1145/3243034
Issue’s Table of Contents

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 26 July 2018
Published in SIGSAM-CCA Volume 52, Issue 1

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