Abstract
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.
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Notes
- 1.
These type of series are often called modified q-analogues of multiple zeta values, since one needs to multiply by \((1-q)^{s_1+\dots +s_l}\) before taking the limit \(q\rightarrow 1\).
- 2.
Recall the series expansion \(\frac{1}{(1-x)^2} = 1 +2 x+ 3 x^2 + 4 x^3+\dots \) and \(\frac{1}{(1-x)^2} \frac{1+x}{1-x} = 1 +4 x+ 9 x^2 + 16 x^3 + \dots \). Now, since we have \(\sum _{k\ge 0} \dim _{\mathbb Q}( M_{k} ( {\text {SL}}_2({\mathbb Z})) \, x^k =\frac{1}{(1-x^4)(1-x^6)} = ( 1+x^4+ x^6 +x^8 +x^{10}+x^{14}) \frac{1}{(1-x^{12})^2}\), the claim follows by replacing x by \(x^{12}\) in the second series expansion.
- 3.
This follows easily from the fact that \(t^{j-1} (1-t)^{s-j}\) with \(j=1,\dots ,s\) (resp. \(j=2,\dots ,s\)) forms a basis of \(\{ Q \in {\mathbb Q}[t] \mid \deg Q \le s-1 \} \) (resp. \(\{ Q \in t {\mathbb Q}[t] \mid \deg Q \le s-1 \} \)).
- 4.
- 5.
Some authors prefer to denote this as the symmetric algebra of a Lie algebra.
- 6.
More precisely, we checked this for a few primes between k and 10007.
References
Bachmann, H.: The algebra of bi-brackets and regularized multiple Eisenstein series. J. Number Theory 200, 260–294 (2019)
Bachmann, H.: Multiple Eisenstein series and \(q\)-analogues of multiple zeta values, In this volume
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)
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)
Bradley, D.M.: Multiple q-zeta values. J. Algebra 283, 752–798 (2005)
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)
Brown, F.: Mixed Tate motives over \({\mathbb{Z}}\). Ann. Math. (2) 175, 949–976 (2012)
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)
Ebrahimi-Fard, K., Manchon, D., Singer, J.: Duality and (q-)multiple zeta values. Adv. Math. 298, 254–285 (2016)
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)
Goncharov, A.B.: Multiple \(\zeta \)-values, Galois groups and geometry of modular varieties. Progr. Math. 201, 361–392 (2001)
Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compositio Math. 142, 307–338 (2006)
Hoffman, M.E.: The algebra of multiple harmonic series. J. Algebra 194, 477–495 (1997)
Hoffman, M.E., Ihara, K.: Quasi-shuffle products revisited. J. Algebra 481, 293–326 (2017)
Ihara, K., Ochiai, H.: Symmetry on linear relations for multiple zeta values. Nagoya Math. J. 189, 49–62 (2008)
Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms, The moduli space of curves. Progr. Math. 129, 165–172 (1995)
Okounkov, A.: Hilbert schemes and multiple \(q\)-zeta values. Funct. Anal. Appl. 48, 138–144 (2014)
Schlesinger, K.: Some remarks on q-deformed multiple polylogarithms. arXiv:math/0111022 [math.QA]
Schneps, L.: ARI, GARI, Zig and Zag: An introduction to Ecalle’s theory of multiple zeta values. arXiv:1507.01534 [math.NT]
Singer, J.: On q-analogues of multiple zeta values. Funct. Approx. Comment. Math. 53, 135–165 (2015)
Takeyama, Y.: The algebra of a q-analogue of multiple harmonic series. SIGMA 9 Paper 061, 1–15 (2013)
Ohno, Y., Okuda, J., Zudilin, W.: Cyclic \(q\)- MZSV sum. J. Number Theory 132, 144–155 (2012)
The PARI Group, PARI/GP version 2.10.0, Univ. Bordeaux (2017). http://pari.math.u-bordeaux.fr/
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)
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)
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)
Zhao, J.: Multiple q-zeta functions and multiple q-polylogarithms. Ramanujan J. 14(2), 189–221 (2007)
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]
Zudilin, W.: Diophantine problems for q-zeta values, (Russian) Mat. Zametki 72(6), 936–940 (2002); translation in Math. Notes 72, 858–862 (2002)
Zudilin, W.: Algebraic relations for multiple zeta values. Russian Math. Surveys 58(1), 1–29 (2003)
Zudilin, W.: Multiple \(q\)-zeta brackets, Mathematics 3:1, special issue Mathematical physics, 119–130 (2015)
Acknowledgements
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.
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Bachmann, H., Kühn, U. (2020). A Dimension Conjecture for 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_9
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