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A Dimension Conjecture for q-Analogues of Multiple Zeta Values

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Periods in Quantum Field Theory and Arithmetic (ICMAT-MZV 2014)

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

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. 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. 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. 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. 4.

    In the articles [1, 2, 4] these spaces were denoted \({{\,\mathrm{\mathscr {BD}}\,}}\) and \({{\,\mathrm{\mathscr {MD}}\,}}\).

  5. 5.

    Some authors prefer to denote this as the symmetric algebra of a Lie algebra.

  6. 6.

    More precisely, we checked this for a few primes between k and 10007.

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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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Authors and Affiliations

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

    Henrik Bachmann

  2. Universität Hamburg Fachbereich Mathematik, Bundesstrasse 55, 20146, Hamburg, Germany

    Ulf Kühn

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  1. Henrik Bachmann
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Correspondence to Henrik Bachmann .

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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

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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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