Multiple Eisenstein Series and q-Analogues of Multiple Zeta Values

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


This work is an example driven overview article of recent works on the connection of multiple zeta values, modular forms and q-analogues of multiple zeta values given by multiple Eisenstein series.


Multiple zeta values q-analogues of multiple zeta values Multiple Eisenstein series Modular forms 



This paper has served as the introductory part of my cumulative thesis written at the University of Hamburg. First of all I would like to thank my supervisor Ulf Kühn for his continuous, encouraging and patient support during the last years. Besides this I also want to thank several people for supporting me during my PhD project by whether giving me suggestion and ideas, letting me give talks on conferences and seminars, proof reading papers or having general discussions on this topic with me. A big “thank you” goes therefore to Olivier Bouillot, Kathrin Bringmann, David Broadhurst, Kurusch Ebrahimi-Fard, Herbert Gangl, José I. Burgos Gil, Masanobu Kaneko, Dominique Manchon, Nils Matthes, Martin Möller, Koji Tasaka, Don Zagier, Jianqiang Zhao and Wadim Zudilin. Finally I would like to thank the referee for various helpful comments and remarks.


  1. 1.
    Andrews, G., Rose, S.: MacMahon’s sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms. J. Reine Angew. Math. 676, 97–103 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bachmann, H.: Multiple Zeta-Werte und die Verbindung zu Modulformen durch multiple Eisensteinreihen. Master thesis, Hamburg University (2012).
  3. 3.
    Bachmann, H.: The algebra of bi-brackets and regularized multiple Eisenstein series. J. Number Theory 200, 260–294 (2019)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bachmann, H.: Multiple Eisenstein series and  \(q\)-analogues of multiple zeta values. Thesis, Hamburg University (2015).
  5. 5.
    Bachmann, H.: Double shuffle relations for q-analogues of multiple zeta values, their derivatives and the connection to multiple Eisenstein series. RIMS Kôyûroku 2017, 22–43 (2015)Google Scholar
  6. 6.
    Bachmann, H., Kühn, U.: The algebra of generating functions for multiple divisor sums and applications to multiple zeta values. Ramanujan J. 40(3), 605–648 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bachmann, H., Kühn, U.: A short note on a conjecture of Okounkov about a  \(q\)-analogue of multiple zeta values. arXiv:1309.3920 [math.NT]
  8. 8.
    Bachmann, H., Kühn, U.: A dimension conjecture for  \(q\)-analogues of multiple zeta values. In This VolumeGoogle Scholar
  9. 9.
    Bachmann, H., Tasaka, K.: The double shuffle relations for multiple Eisenstein series. Nagoya Math. J. 230, 1–33 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bachmann, H., Tsumura, H.: Multiple series of Eisenstein type. Ramanujan J. 42(2), 479–489 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Borwein, J., Bradley, D.: Thirty-two Goldbach variations. Int. J. Number Theory 02, 65–103 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bouillot, O.: The algebra of multitangent functions. J. Algebra 410, 148–238 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bouillot, O.: Table of reduction of multitangent functions of weight up to 10 (2012).
  14. 14.
    Bradley, D.M.: Multiple q-zeta values. J. Algebra 283, 752–798 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    Ecalle, J.: The flexion structure and dimorphy: flexion units, singulators, generators, and the enumeration of multizeta irreducibles. In: Asymptotics in Dynamics, Geometry and PDEs, Generalized Borel Summation, vol. II, pp. 27–211 (2011)Google Scholar
  17. 17.
    Ebrahimi-Fard, K., Manchon, D., Singer, J.: Duality and (q-)multiple zeta values. Adv. Math. 298, 254–285 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ebrahimi-Fard, K., Manchon, D., Medina, J.C.: Unfolding the double shuffle structure of q-multiple zeta values. Bull. Austral. Math. Soc. 91(3), 368–388 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gangl, H., Kaneko, M., Zagier, D.: Double zeta values and modular forms. Automorphic Forms and Zeta Functions, pp. 71–106. World Science Publisher, Hackensack, NJ (2006)CrossRefGoogle Scholar
  20. 20.
    Goncharov, A.B.: Galois symmetries of fundamental groupoids and noncommutative geometry. Duke Math. J. 128(2), 209–284 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hoffman, M.E.: Quasi-shuffle products. J. Algebraic Combin. 11(1), 49–68 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hoffman, M.E., Ihara, K.: Quasi-shuffle products revisited. J. Algebra 481, 293–326 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ihara, K.: Derivation and double shuffle relations for multiple zeta values, joint work with M. Kaneko, D. Zagier. RIMS Kôyûroku 1549, 47–63Google Scholar
  24. 24.
    Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142, 307–338 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ihara, K., Ochiai, H.: Symmetry on linear relations for multiple zeta values. Nagoya Math. J. 189, 49–62 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kaneko, M., Tasaka, K.: Double zeta values, double Eisenstein series, and modular forms of level 2. Math. Ann. 357(3), 1091–1118 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Okounkov, A.: Hilbert schemes and multiple \(q\)-zeta values. Funct. Anal. Appl. 48, 138–144 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Ohno, Y., Okuda, J., Zudilin, W.: Cyclic \(q\)-MZSV sum. J. Number Theory 132(1), 144–155 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Qin, Z., Yu, F.: On Okounkov’s conjecture connecting Hilbert schemes of points and multiple q-zeta values. Int. Math. Res. Not. 2, 321–361 (2018)Google Scholar
  30. 30.
    Rose, S.: Quasi-modularity of generalized sum-of-divisors functions. Res. Number Theory 1, Art. 18, 11 pp (2015)Google Scholar
  31. 31.
    Schlesinger, K.-G.: Some remarks on q-deformed multiple polylogarithms. arXiv:math/0111022 [math.QA]
  32. 32.
    Singer, J.: On q-analogues of multiple zeta values. Funct. Approx. Comment. Math. 53(1), 135–165 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Takeyama, Y.: The algebra of a q-analogue of multiple harmonic series. SIGMA Symmetry Integrability Geom. Methods Appl. 9, Paper 061 (2013)Google Scholar
  34. 34.
    Yuan, H., Zhao, J.: Double shuffle relations of double zeta values and double Eisenstein series of level N. J. Lond. Math. Soc. (2) 92(3), 520–546 (2015)Google Scholar
  35. 35.
    Yuan, H., Zhao, J.: Multiple Divisor Functions and Multiple Zeta Values at Level N. arXiv:1408.4983 [math.NT]
  36. 36.
    Zagier, D.: Elliptic modular forms and their applications. The 1-2-3 of Modular Forms, pp. 1–103. Universitext Springer, Berlin (2008)Google Scholar
  37. 37.
    Zagier, D.: Periods of modular forms, traces of Hecke operators, and multiple zeta values. RIMS Kôyûroku 843, 162–170 (1993)MathSciNetGoogle Scholar
  38. 38.
    Zhao, J.: Multiple q-zeta functions and multiple q-polylogarithms. Ramanujan J. 14(2), 189–221 (2007)MathSciNetCrossRefGoogle Scholar
  39. 39.
    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]
  40. 40.
    Zorich, A.: Flat surfaces. Frontiers in Number Theory, Physics, and Geometry, vol. I, Springer (2006)Google Scholar
  41. 41.
    Zudilin, W.: Multiple  \(q\)-zeta brackets. Math. 3:1, Spec. Issue Math. Phys. 119–130 (2015)Google Scholar
  42. 42.
    Zudilin, W.: Algebraic relations for multiple zeta values, (Russian. Russian summary) Uspekhi Mat. Nauk 58 (2003)Google Scholar

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Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan

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