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.
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Notes
- 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.
- 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.
We set \([s_1,\dots ,s_l] =1\) for \(l=0\).
- 5.
- 6.
Here \(\delta _{a,b}\) denotes the Kronecker delta, i.e \(\delta _{a,b}\) is 1 for \(a=b\) and 0 otherwise.
- 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.
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.
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.
- 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 \(\genfrac[]{0.0pt}{}{2,1}{1,0} \in \ker Z_4\).
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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.
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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
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