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Uspekhi Mat. Nauk, 2003, Volume 58, Issue 1(349), Pages 3–32 (Mi umn592)  

This article is cited in 46 scientific papers (total in 46 papers)

Algebraic relations for multiple zeta values

W. V. Zudilin

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

Abstract: The survey is devoted to the multidimensional generalization of the Riemann zeta function as a function of a positive integral argument.

DOI: https://doi.org/10.4213/rm592

Full text: PDF file (402 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2003, 58:1, 1–29

Bibliographic databases:

Document Type: Article
UDC: 511.3+512.573
MSC: Primary 11M06; Secondary 11G55, 16W25
Received: 30.10.2001

Citation: W. V. Zudilin, “Algebraic relations for multiple zeta values”, Uspekhi Mat. Nauk, 58:1(349) (2003), 3–32; Russian Math. Surveys, 58:1 (2003), 1–29

Citation in format AMSBIB
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    Erratum

    This publication is cited in the following articles:
    1. Blümlein J., “Reduction of multiple harmonic sums and harmonic polylogarithms”, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 534:1-2 (2004), 279–283  crossref  adsnasa  isi  elib
    2. Blumlein J., “Algebraic relations between harmonic sums and associated quantities”, Comput. Phys. Comm., 159:1 (2004), 19–54  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Hoffman M.E., “The Hopf algebra structure of multiple harmonic sums”, Nuclear Phys. B Proc. Suppl., 135 (2004), 215–219  crossref  mathscinet  adsnasa  isi
    4. Blumlein J., “Mathematical structure of anomalous dimensions and QCD Wilson coefficients in higher order”, Nuclear Phys. B Proc. Suppl., 135 (2004), 225–231  crossref  mathscinet  adsnasa  isi
    5. M. Waldschmidt, “Open Diophantine problems”, Mosc. Math. J., 4:1 (2004), 245–305  mathnet  mathscinet  zmath
    6. Bradley D.A., “Duality for finite multiple harmonic $q$-series”, Discrete Math., 300:1-3 (2005), 44–56  crossref  mathscinet  zmath  isi
    7. Bradley D.M., “Multiple $q$-zeta values”, J. Algebra, 283:2 (2005), 752–798  crossref  mathscinet  zmath  isi  elib
    8. Hoffman M.E., “Algebraic aspects of multiple zeta values”, Zeta Functions, Topology and Quantum Physics, Developments in Mathematics, 14, 2005, 51–73  crossref  mathscinet  zmath  isi
    9. Waldschmidt M., “Hopf algebras and transcendental numbers”, Zeta Functions, Topology and Quantum Physics, Developments in Mathematics, 14, 2005, 197–219  crossref  mathscinet  zmath  isi
    10. Yu. V. Nesterenko, “On an Identity of Mahler”, Math. Notes, 79:1 (2006), 97–108  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. Borwein J.M., Bradley D.M., “Thirty-two goldbach variations”, Int. J. Number Theory, 2:1 (2006), 65–103  crossref  mathscinet  zmath  isi
    12. Murty M.R., Sinha K., “Multiple Hurwitz zeta functions”, Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, Proceedings of Symposia in Pure Mathematics, 75, 2006, 135–156  crossref  mathscinet  zmath  isi
    13. S. A. Zlobin, “Rhin Integrals”, Math. Notes, 81:2 (2007), 201–212  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    14. Bradley D.M., “On the sum formula for multiple $q$-zeta values”, Rocky Mountain J. Math., 37:5 (2007), 1427–1434  crossref  mathscinet  zmath  isi
    15. Cariñena J.F., Ebrahimi-Fard K., Figueroa H., Gracia-Bondía J.M., “Hopf algebras in dynamical systems theory”, Int. J. Geom. Methods Mod. Phys., 4:4 (2007), 577–646  crossref  mathscinet  zmath  isi
    16. Kalmykov M.Yu., Warda B.F.L., Yost S.A., “Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter”, Journal of High Energy Physics, 2007, no. 10, 048  crossref  mathscinet  isi
    17. Fischler S., “Multiple series connected to Hoffman's conjecture on multiple zeta values”, J. Algebra, 320:4 (2008), 1682–1703  crossref  mathscinet  zmath  isi  elib
    18. Zhao Jianqiang, “Renormalization of multiple $q$-zeta values”, Acta Math. Sin. (Engl. Ser.), 24:10 (2008), 1593–1616  crossref  mathscinet  zmath  isi
    19. Ohno Ya., Zudilin W., “Zeta stars”, Communications in Number Theory and Physics, 2:2 (2008), 325–347  crossref  mathscinet  zmath  isi
    20. Guo L., Zhang B., “Differential Algebraic Birkhoff Decomposition and the renormalization of multiple zeta values”, Journal of Number Theory, 128:8 (2008), 2318–2339  crossref  mathscinet  zmath  isi  elib
    21. Bluemlein J., Broadhurst D.J., Vermaseren J.A.M., “The Multiple Zeta Value data mine”, Comput. Phys. Comm., 181:3 (2010), 582–625  crossref  mathscinet  zmath  adsnasa  isi  elib
    22. Manchon D., Paycha S., “Nested Sums of Symbols and Renormalized Multiple Zeta Values”, International Mathematics Research Notices, 2010, no. 24, 4628–4697  crossref  mathscinet  zmath  isi  elib
    23. Guo Li, Zhang Bin, “Polylogarithms and multiple zeta values from free Rota-Baxter algebras”, Science China-Mathematics, 53:9 (2010), 2239–2258  crossref  mathscinet  zmath  adsnasa  isi
    24. S. A. Zlobin, “Special values of generalized polylogarithms”, J. Math. Sci., 182:4 (2012), 484–504  mathnet  crossref  mathscinet
    25. Ohno Ya., Okuda J.-i., Zudilin W., “Cyclic q-MZSV sum”, J Number Theory, 132:1 (2012), 144–155  crossref  mathscinet  zmath  isi  elib
    26. Dunin-Barkowski P., Sleptsov A., Smirnov A., “Explicit Computation of the Drinfeld Associator in the Case of the Fundamental Representation of Gl(N)”, J. Phys. A-Math. Theor., 45:38 (2012), 385204  crossref  mathscinet  zmath  adsnasa  isi  elib
    27. Leurent S., Volin D., “Multiple Zeta Functions and Double Wrapping in Planar N=4 SYM”, Nucl. Phys. B, 875:3 (2013), 757–789  crossref  mathscinet  zmath  adsnasa  isi  elib
    28. Olivier Bouillot, “The algebra of multitangent functions”, Journal of Algebra, 410 (2014), 148  crossref  mathscinet  zmath  isi
    29. A. Yu. Okounkov, “Hilbert Schemes and Multiple $q$-Zeta Values”, Funct. Anal. Appl., 48:2 (2014), 138–144  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    30. Olivier Bouillot, “The algebra of Hurwitz multizeta functions”, Comptes Rendus Mathematique, 2014  crossref  mathscinet  isi
    31. J.C.astillo Medina, Kurusch Ebrahimi-Fard, Dominique Manchon, “On Euler’s decomposition formula for
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      q MZVs”, Ramanujan J, 2014  crossref  mathscinet  isi
    32. Khodabakhsh Hessami Pilehrood, Tatiana Hessami Pilehrood, “On
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      q -analogues of two-one formulas for multiple harmonic sums and multiple zeta star values”, Monatsh Math, 2014  crossref  mathscinet  isi
    33. JAIME CASTILLO-MEDINA, KURUSCH EBRAHIMI-FARD, DOMINIQUE MANCHON, “UNFOLDING THE DOUBLE SHUFFLE STRUCTURE OF -MULTIPLE ZETA VALUES”, Bull. Aust. Math. Soc, 91:03 (2015), 368  crossref  mathscinet  zmath  isi
    34. Bouillot O., “on Hurwitz Multizeta Functions”, 71, 2015, 68–124  crossref  mathscinet  zmath  isi
    35. Zudilin W., “Multiple Q-Zeta Brackets”, 3, no. 1, 2015, 119–130  crossref  zmath  isi
    36. Ebrahimi-Fard K., Manchon D., Singer J., “Renormalisation of Q-Regularised Multiple Zeta Values”, 106, no. 3, 2016, 365–380  crossref  mathscinet  zmath  isi
    37. Ebrahimi-Fard K., Manchon D., Singer J., “Duality and ( q -)multiple zeta values”, Adv. Math., 298 (2016), 254–285  crossref  mathscinet  zmath  isi  scopus
    38. Gencev M., “On restricted sum formulas for multiple zeta values with even arguments”, Arch. Math., 107:1 (2016), 9–22  crossref  mathscinet  zmath  isi  elib  scopus
    39. Singer J., “On Bradley's q -MZVs and a generalized Euler decomposition formula”, J. Algebra, 454 (2016), 92–122  crossref  mathscinet  zmath  isi  scopus
    40. D'Hoker E., Kaidi J., “Hierarchy of modular graph identities”, J. High Energy Phys., 2016, no. 11, 051  crossref  mathscinet  isi  scopus
    41. Saito Sh., Wakabayashi N., “Bowman-Bradley type theorem for finite multiple zeta values”, Tohoku Math. J., 68:2 (2016), 241–251  crossref  mathscinet  zmath  isi
    42. Ch. Berg, R. Szwarc, “Symmetric moment problems and a conjecture of Valent”, Sb. Math., 208:3 (2017), 335–359  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    43. Jian R.-Q., “Quantum quasi-shuffle algebras II”, J. Algebra, 472 (2017), 480–506  crossref  mathscinet  zmath  isi  scopus
    44. Qin Zh., Yu F., “On Okounkov'S Conjecture Connecting Hilbert Schemes of Points and Multiple Q-Zeta Values”, Int. Math. Res. Notices, 2018, no. 2, 321–361  crossref  mathscinet  isi
    45. Ebrahimi-Fard K., Manchon D., Singer J., Zhao J., “Renormalisation Group For Multiple Zeta Values”, Commun. Number Theory Phys., 12:1 (2018), 75–96  crossref  mathscinet  zmath  isi
    46. D'Hoker E., Duke W., “Fourier Series of Modular Graph Functions”, J. Number Theory, 192 (2018), 1–36  crossref  mathscinet  zmath  isi  scopus
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