Skip to main content
SpringerLink
Log in

Multiple Eisenstein Series and q-Analogues of Multiple Zeta Values

  • Conference paper
  • First Online:
Book cover Periods in Quantum Field Theory and Arithmetic (ICMAT-MZV 2014)

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

  • 599 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Some authors use the opposite convention \(0< n_1<\dots < n_l\) in the definition of multiple zeta values. This is in particular the case for the work [9], where this opposite convention is used for multiple zeta values and multiple Eisenstein series.

  2. 2.

    Further, one can prove the relation \(\zeta (3) = \zeta (2,1)\) between multiple zeta values by multiplying both sides in (20) with \((1-q)^3\) and then take the limit \(q\rightarrow 1\). We will discuss this in Sect. 7.

  3. 3.

    For convenience we recall that the Bernoulli numbers \(B_k\) are defined by \(\frac{X}{e^X-1} =: \sum _{k\ge 0} \frac{B_k}{k!}X^k\).

  4. 4.

    We set \([s_1,\dots ,s_l] =1\) for \(l=0\).

  5. 5.

    The bi-brackets and their generating series also give examples of what is called a bimould by Ecalle in [16]. In his language the partition relation (23) states that the bimould of generating series of bi-brackets is swap invariant.

  6. 6.

    Here \(\delta _{a,b}\) denotes the Kronecker delta, i.e \(\delta _{a,b}\) is 1 for \(a=b\) and 0 otherwise.

  7. 7.

    If one likes to interpret the integrals as real integrals, then the passage from \(\mathscr {I}\) to \(\mathscr {I}^1\) regularizes these integrals such that “\(-\log (0) = \int _{1> t > 0} \frac{dt}{t} := 0\)”.

  8. 8.

    This notion fits well with the iterated integral expression of multiple zeta values. Recall that

    $$\begin{aligned} \zeta (2,3) = \int _{{\small 1> t_1> \dots> t_5 > 0}} \underbrace{\frac{dt_1}{t_1} \cdot \frac{dt_2}{1-t_2}}_{2} \cdot \underbrace{ \frac{dt_3}{t_3}\cdot \frac{dt_4}{t_4} \cdot \frac{dt_5}{1-t_5}}_{3} \,. \end{aligned}$$

    This corresponds to I(2, 3) (but is of course not the same since the I are formal symbols).

  9. 9.

    In [24] the authors introduced the notion of extended double shuffle relations. We use this notion here for smaller subset of these relations given there as the relations described in statement (3) on page 315.

  10. 10.

    This map is similar to the evaluation map \(Z^*: \mathfrak H^1 \rightarrow \mathbb {R}[T]\), of stuffle regularized multiple zeta values, given in Proposition 1 in [24]. We used this map in the previous sections (Proposition 3.2) with \(T=0\).

  11. 11.

    That the last term \(\genfrac[]{0.0pt}{}{2,1}{1,0}\) in (39) is in the kernel of \(Z_4\) can be proven in the following way: In Proposition 7.2 [6] it is shown, that an element \(f = \sum _{n>0} a_n q^n\) with \(a_n = O(n^m)\) and \(m<k-1\) is in the kernel of \(Z_k\). Here we have

    $$\begin{aligned} \genfrac[]{0.0pt}{}{2,1}{1,0} = \sum _{\begin{array}{c} u_1> u_2> 0\\ v_1 , v_2> 0 \end{array}} v_1 u_1 q^{v_1 u_1 + v_2 u_2} < \sum _{\begin{array}{c} u_1,u_1 0\\ v_1 , v_2 > 0 \end{array}} v_1 u_1 q^{v_1 u_1 + v_2 u_2} = {\text {d}}[1] \cdot [1]\,, \end{aligned}$$

    where the < is meant to be coefficient wise. Since the coefficients of \( {\text {d}}[1] \cdot [1]\) grow like \(n^2 \log (n)^2\) we conclude \(\genfrac[]{0.0pt}{}{2,1}{1,0} \in \ker Z_4\).

References

  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)

    MathSciNet  MATH  Google Scholar 

  2. Bachmann, H.: Multiple Zeta-Werte und die Verbindung zu Modulformen durch multiple Eisensteinreihen. Master thesis, Hamburg University (2012). http://www.henrikbachmann.com

  3. Bachmann, H.: The algebra of bi-brackets and regularized multiple Eisenstein series. J. Number Theory 200, 260–294 (2019)

    Article  MathSciNet  Google Scholar 

  4. Bachmann, H.: Multiple Eisenstein series and  \(q\)-analogues of multiple zeta values. Thesis, Hamburg University (2015). http://www.henrikbachmann.com

  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. 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)

    Article  MathSciNet  Google Scholar 

  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. Bachmann, H., Kühn, U.: A dimension conjecture for  \(q\)-analogues of multiple zeta values. In This Volume

    Google Scholar 

  9. Bachmann, H., Tasaka, K.: The double shuffle relations for multiple Eisenstein series. Nagoya Math. J. 230, 1–33 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Bachmann, H., Tsumura, H.: Multiple series of Eisenstein type. Ramanujan J. 42(2), 479–489 (2017)

    Article  MathSciNet  Google Scholar 

  11. Borwein, J., Bradley, D.: Thirty-two Goldbach variations. Int. J. Number Theory 02, 65–103 (2006)

    Article  MathSciNet  Google Scholar 

  12. Bouillot, O.: The algebra of multitangent functions. J. Algebra 410, 148–238 (2014)

    Article  MathSciNet  Google Scholar 

  13. Bouillot, O.: Table of reduction of multitangent functions of weight up to 10 (2012). http://www-igm.univ-mlv.fr/~bouillot/Tables_de_multitangentes.pdf

  14. Bradley, D.M.: Multiple q-zeta values. J. Algebra 283, 752–798 (2005)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Ebrahimi-Fard, K., Manchon, D., Singer, J.: Duality and (q-)multiple zeta values. Adv. Math. 298, 254–285 (2016)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  20. Goncharov, A.B.: Galois symmetries of fundamental groupoids and noncommutative geometry. Duke Math. J. 128(2), 209–284 (2005)

    Article  MathSciNet  Google Scholar 

  21. Hoffman, M.E.: Quasi-shuffle products. J. Algebraic Combin. 11(1), 49–68 (2000)

    Article  MathSciNet  Google Scholar 

  22. Hoffman, M.E., Ihara, K.: Quasi-shuffle products revisited. J. Algebra 481, 293–326 (2017)

    Article  MathSciNet  Google Scholar 

  23. Ihara, K.: Derivation and double shuffle relations for multiple zeta values, joint work with M. Kaneko, D. Zagier. RIMS Kôyûroku 1549, 47–63

    Google Scholar 

  24. Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142, 307–338 (2006)

    Article  MathSciNet  Google Scholar 

  25. Ihara, K., Ochiai, H.: Symmetry on linear relations for multiple zeta values. Nagoya Math. J. 189, 49–62 (2008)

    Article  MathSciNet  Google Scholar 

  26. Kaneko, M., Tasaka, K.: Double zeta values, double Eisenstein series, and modular forms of level 2. Math. Ann. 357(3), 1091–1118 (2013)

    Article  MathSciNet  Google Scholar 

  27. Okounkov, A.: Hilbert schemes and multiple \(q\)-zeta values. Funct. Anal. Appl. 48, 138–144 (2014)

    Article  MathSciNet  Google Scholar 

  28. Ohno, Y., Okuda, J., Zudilin, W.: Cyclic \(q\)-MZSV sum. J. Number Theory 132(1), 144–155 (2012)

    Article  MathSciNet  Google Scholar 

  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. Rose, S.: Quasi-modularity of generalized sum-of-divisors functions. Res. Number Theory 1, Art. 18, 11 pp (2015)

    Google Scholar 

  31. Schlesinger, K.-G.: Some remarks on q-deformed multiple polylogarithms. arXiv:math/0111022 [math.QA]

  32. Singer, J.: On q-analogues of multiple zeta values. Funct. Approx. Comment. Math. 53(1), 135–165 (2015)

    Article  MathSciNet  Google Scholar 

  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. 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. Yuan, H., Zhao, J.: Multiple Divisor Functions and Multiple Zeta Values at Level N. arXiv:1408.4983 [math.NT]

  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. Zagier, D.: Periods of modular forms, traces of Hecke operators, and multiple zeta values. RIMS Kôyûroku 843, 162–170 (1993)

    MathSciNet  Google Scholar 

  38. Zhao, J.: Multiple q-zeta functions and multiple q-polylogarithms. Ramanujan J. 14(2), 189–221 (2007)

    Article  MathSciNet  Google Scholar 

  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. Zorich, A.: Flat surfaces. Frontiers in Number Theory, Physics, and Geometry, vol. I, Springer (2006)

    Google Scholar 

  41. Zudilin, W.: Multiple  \(q\)-zeta brackets. Math. 3:1, Spec. Issue Math. Phys. 119–130 (2015)

    Google Scholar 

  42. Zudilin, W.: Algebraic relations for multiple zeta values, (Russian. Russian summary) Uspekhi Mat. Nauk 58 (2003)

    Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

    Henrik Bachmann

Authors
  1. Henrik Bachmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Henrik Bachmann .

Editor information

Editors and Affiliations

  1. Institute of Mathematical Sciences (ICMAT), Spanish National Research Council (CSIC), Madrid, Spain

    José Ignacio Burgos Gil

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  3. Department of Mathematical Sciences, Durham University, Durham, UK

    Herbert Gangl

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bachmann, H. (2020). Multiple Eisenstein Series and q-Analogues of Multiple Zeta Values. In: Burgos Gil, J., Ebrahimi-Fard, K., Gangl, H. (eds) Periods in Quantum Field Theory and Arithmetic. ICMAT-MZV 2014. Springer Proceedings in Mathematics & Statistics, vol 314. Springer, Cham. https://doi.org/10.1007/978-3-030-37031-2_8

Download citation

Publish with us

Policies and ethics

Search

Navigation