A Dimension Conjecture for q-Analogues of Multiple Zeta Values

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 314)


We study a class of q-analogues of multiple zeta values given by certain formal q-series with rational coefficients. After introducing a notion of weight and depth for these q-analogues of multiple zeta values we present dimension conjectures for the spaces of their weight- and depth-graded parts, which have a similar shape as the conjectures of Zagier and Broadhurst-Kreimer for multiple zeta values.


Multiple zeta values q-Analogues of multiple zeta values Modular forms Dimension conjecture 



We would like to thank N. Matthes and the referees for their careful reading of our manuscript and their valuable comments. The first author would also like to thank the Max-Planck Institute for Mathematics in Bonn for hospitality and support.


  1. 1.
    Bachmann, H.: The algebra of bi-brackets and regularized multiple Eisenstein series. J. Number Theory 200, 260–294 (2019)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bachmann, H.: Multiple Eisenstein series and \(q\)-analogues of multiple zeta values, In this volumeGoogle Scholar
  3. 3.
    Bachmann, H.: Double shuffle relations for q-analogues of multiple zeta values, their derivatives and the connection to multiple Eisenstein series. RIMS Kôyûroku No. 2017, 22–43 (2015)Google Scholar
  4. 4.
    Bachmann, H., Kühn, U.: The algebra of generating functions for multiple divisor sums and applications to multiple zeta values. Ramanujan J. 40, 605–648 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bradley, D.M.: Multiple q-zeta values. J. Algebra 283, 752–798 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Broadhurst, D., Kreimer, D.: Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. Phys. Lett. B 393, 403–412 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brown, F.: Mixed Tate motives over \({\mathbb{Z}}\). Ann. Math. (2) 175, 949–976 (2012)Google Scholar
  8. 8.
    Ecalle, J.: The flexion structure and dimorphy: flexion units, singulators, generators, and the enumeration of multizeta irreducibles. Asymptotics in dynamics, geometry and PDEs, generalized Borel summation II, 27–211 (2011)Google Scholar
  9. 9.
    Ebrahimi-Fard, K., Manchon, D., Singer, J.: Duality and (q-)multiple zeta values. Adv. Math. 298, 254–285 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Foata, D.: Eulerian polynomials: from Euler’s Time to the Present, The legacy of Alladi Ramakrishnan in the mathematical sciences, pp. 253–273. Springer, New York (2010)Google Scholar
  11. 11.
    Goncharov, A.B.: Multiple \(\zeta \)-values, Galois groups and geometry of modular varieties. Progr. Math. 201, 361–392 (2001)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compositio Math. 142, 307–338 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hoffman, M.E.: The algebra of multiple harmonic series. J. Algebra 194, 477–495 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hoffman, M.E., Ihara, K.: Quasi-shuffle products revisited. J. Algebra 481, 293–326 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ihara, K., Ochiai, H.: Symmetry on linear relations for multiple zeta values. Nagoya Math. J. 189, 49–62 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms, The moduli space of curves. Progr. Math. 129, 165–172 (1995)zbMATHGoogle Scholar
  17. 17.
    Okounkov, A.: Hilbert schemes and multiple \(q\)-zeta values. Funct. Anal. Appl. 48, 138–144 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Schlesinger, K.: Some remarks on q-deformed multiple polylogarithms. arXiv:math/0111022 [math.QA]
  19. 19.
    Schneps, L.: ARI, GARI, Zig and Zag: An introduction to Ecalle’s theory of multiple zeta values. arXiv:1507.01534 [math.NT]
  20. 20.
    Singer, J.: On q-analogues of multiple zeta values. Funct. Approx. Comment. Math. 53, 135–165 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Takeyama, Y.: The algebra of a q-analogue of multiple harmonic series. SIGMA 9 Paper 061, 1–15 (2013)Google Scholar
  22. 22.
    Ohno, Y., Okuda, J., Zudilin, W.: Cyclic \(q\)- MZSV sum. J. Number Theory 132, 144–155 (2012)Google Scholar
  23. 23.
    The PARI Group, PARI/GP version 2.10.0, Univ. Bordeaux (2017).
  24. 24.
    Pupyrev, Y.: On the linear and algebraic independence of q-zeta values, (Russian. Russian summary) Mat. Zametki 78(4), 608–613 (2005); translation in Math. Notes 78(3–4), 563–568 (2005)Google Scholar
  25. 25.
    Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. Modular functions of one variable VI, Lecture Notes in Math. 627, Springer, Berlin, 105–169 (1977)Google Scholar
  26. 26.
    Zagier, D.: Values of zeta functions and their applications. First European Congress of Mathematics, Volume II, Progress in Math. 120, Birkhäuser-Verlag, Basel, 497–512 (1994)Google Scholar
  27. 27.
    Zhao, J.: Multiple q-zeta functions and multiple q-polylogarithms. Ramanujan J. 14(2), 189–221 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zhao, J.: Uniform approach to double shuffle and duality relations of various q-analogs of multiple zeta values via Rota-Baxter algebras. arXiv:1412.8044 [math.NT]
  29. 29.
    Zudilin, W.: Diophantine problems for q-zeta values, (Russian) Mat. Zametki 72(6), 936–940 (2002); translation in Math. Notes 72, 858–862 (2002)Google Scholar
  30. 30.
    Zudilin, W.: Algebraic relations for multiple zeta values. Russian Math. Surveys 58(1), 1–29 (2003)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Zudilin, W.: Multiple \(q\)-zeta brackets, Mathematics 3:1, special issue Mathematical physics, 119–130 (2015)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan
  2. 2.Universität Hamburg Fachbereich MathematikHamburgGermany

Personalised recommendations