The algebra of generating functions for multiple divisor sums and applications to multiple zeta values. (English) Zbl 1412.11101
Summary: We study the algebra \(\mathcal{MD}\) of generating functions for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in \(q\) with coefficients in \(\mathbb {Q}\) arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra \(\mathcal{MD}\) is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in \(\mathcal{MD}\). The (quasi-)modular forms for the full modular group \(\mathrm{SL}_2(\mathbb {Z})\) constitute a subalgebra of \(\mathcal{MD}\), and this also yields linear relations in \(\mathcal{MD}\). Generating functions of multiple divisor sums can be seen as a \(q\)-analogue of multiple zeta values. Studying a certain map from this algebra into the real numbers we will derive a new explanation for relations between multiple zeta values,
including those of length 2, coming from modular forms.
MSC:
| 11M32 | Multiple Dirichlet series and zeta functions and multizeta values |
| 11F11 | Holomorphic modular forms of integral weight |
| 13J05 | Power series rings |
| 33E20 | Other functions defined by series and integrals |
| 05A30 | \(q\)-calculus and related topics |
Keywords:
multiple zeta values; \(q\)-analogues of multiple zeta values; (quasi-)modular forms; multiple divisor sums; quasi-shuffle algebras; multiple Eisenstein seriesReferences:
| [1] | Andrews, G., Rose, S.: MacMahon’s sum-of-divisors functions, Chebyshev polynomials, and quasi-modular forms. Preprint arXiv:1010.5769 [math.NT] · Zbl 1337.11002 |
| [2] | Bachmann, H.: Multiple Zeta-werte und die Verbindung zu Modulformen durch Multiple Eisensteinreihen. Master Thesis, Universität Hamburg (2012) · Zbl 1139.11322 |
| [3] | Bachmann, H.: The algebra of bi-brackets and regularised multiple Eisenstein series. Preprint arXiv:1504.08138 [math.NT] · Zbl 1443.11174 |
| [4] | Bachmann, H.: Multiple Eisenstein series and \[q\] q-analogues of multiple zeta values. PhD Thesis, Universität Hamburg (in press) · Zbl 1455.11123 |
| [5] | Bachmann, H., Tasaka, K.: The double shuffle relations for multiple Eisenstein series. Preprint arXiv:1501.03408 [math.NT] · Zbl 1453.11115 |
| [6] | Bachmann, H., Kühn, U.: A short note on a conjecture of Okounkov about a \[q\] q-analogue of multiple zeta values. Preprint arXiv:1407.6796 [math.NT] · Zbl 1444.81021 |
| [7] | Bradley, D.: Multiple q-zeta values. J. Algebra 283(2), 752-798 (2005) · Zbl 1114.11075 · doi:10.1016/j.jalgebra.2004.09.017 |
| [8] | Costin, O., Garoufalidis, S.: Resurgence of the fractional polylogarithms. Math. Res. Lett. 16(Nr. 5), 817-826 (2009) · Zbl 1201.30044 · doi:10.4310/MRL.2009.v16.n5.a5 |
| [9] | Foata, D.: Eulerian polynomials: From Euler’s Time to the Present. The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, vol. 253-273. Springer, New York (2010) · Zbl 1322.01007 |
| [10] | Gangl, H.; Kaneko, M.; Zagier, D., Double zeta values and modular forms, 71-106 (2006), Hackensack · Zbl 1122.11057 · doi:10.1142/9789812774415_0004 |
| [11] | Hoffman, M.: The Algebra of Multiple Harmonic Series. J. Algebra 194(2), 477-495 (1997). doi:10.1006/jabr.1997.7127 · Zbl 0881.11067 · doi:10.1006/jabr.1997.7127 |
| [12] | Hoffman, M., Ihara, K.: Quasi-shuffle products revisited, Max-Planck-Institut für Mathematik Preprint Series (16) (2012) · Zbl 1378.16042 |
| [13] | Hoffman, M., Ohno, Y.: Relations of multiple zeta values and their algebraic expression. J. Algebra 262(2), 332-347 (2003) · Zbl 1139.11322 · doi:10.1016/S0021-8693(03)00016-4 |
| [14] | Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142, 307-338 (2006) · Zbl 1186.11053 · doi:10.1112/S0010437X0500182X |
| [15] | Kaneko, M., Kurokawa, N., Wakayama, M.: A variation of Euler’s approach to values of the Riemann zeta function. Kyushu J. Math. 57, 175-192 (2003) · Zbl 1067.11053 · doi:10.2206/kyushujm.57.175 |
| [16] | MacMahon, P.A.: In: Andrews, G. (ed.) Divisors of Numbers and Their Continuations in the Theory of Partitions. MIT Press, Cambridge (1986). Reprinted: Percy A. MacMahon Collected Papers |
| [17] | Ohno, Y., Okuda, J., Zudilin, W.: Cyclic q-MZSV sum. J. Number Theory 132, 144-155 (2012) · Zbl 1268.11119 · doi:10.1016/j.jnt.2011.08.001 |
| [18] | Ohno, Y., Zagier, D.: Multiple zeta values of fixed weight, depth, and height. Indag. Math. (N.S.) 12(4), 483-487 (2001) · Zbl 1031.11053 · doi:10.1016/S0019-3577(01)80037-9 |
| [19] | Okuda, J., Takeyama, Y.: On relations for the multiple q-zeta values. Ramanujan J. 14, 379-387 (2007) · Zbl 1211.11099 · doi:10.1007/s11139-007-9053-5 |
| [20] | Pupyrev, Yu. A.: Linear and algebraic independence of q-zeta values. Math. Notes 78(4), 563-568 (2005). Translated from Matematicheskie Zametki, vol. 78, no. 4, 2005, pp. 608-613 · Zbl 1160.11338 |
| [21] | Takeyama, Y.: The algebra of a q-analogue of multiple harmonic series. Preprint arXiv:1306.6164 [math.NT] · Zbl 1286.11135 |
| [22] | Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In: Modular Functions of One Variable, VI (Proceedings of Second International Conference, University of Bonn, Bonn, 1976), pp. 105-169. Lecture Notes in Mathematics, vol. 627. Springer, Berlin (1977) · Zbl 0372.10017 |
| [23] | Zagier, D., Elliptic modular forms and their applications, 1-103 (2008), Berlin · Zbl 1259.11042 |
| [24] | Zhao, J.: Multiple q-zeta functions and multiple q-polylogarithms. Ramanujan J. 14, 189-221 (2007) · Zbl 1200.11065 · doi:10.1007/s11139-007-9025-9 |
| [25] | Zudilin, V.V.: Diophantine problems for q-zeta values. Math. Notes 72(6), 858-862 (2002). Translated from Matematicheskie Zametki, vol. 72, no. 6, 2002, pp. 936-940 · Zbl 1044.11066 · doi:10.1023/A:1021450231834 |
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