Partitions, multiple zeta values and the q-bracket

• Henrik Bachmann ^[1] ; Jan-Willem van Ittersum ^[2]
  1. [1] Nagoya University

     Nagoya University

     Naka-ku, Japón

     
  2. [2] Utrecht University

     Utrecht University

     Países Bajos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 1, 2024, 46 págs.
• Idioma: inglés
• DOI: 10.1007/s00029-023-00893-4
• Enlaces
  • Texto completo
• Resumen

  • We provide a framework for relating certain q-series defined by sums over partitions to multiple zeta values. In particular, we introduce a space of polynomial functions on partitions for which the associated q-series are q-analogues of multiple zeta values. By explicitly describing the (regularized) multiple zeta values one obtains as q → 1, we extend previous results known in this area. Using this together with the fact that other families of functions on partitions, such as shifted symmetric functions, are elements in our space will then give relations among (q-analogues of) multiple zeta values. Conversely, we will show that relations among multiple zeta values can be ‘lifted’ to the world of functions on partitions, which provides new examples of functions for which the associated q-series are quasimodular.

• Referencias bibliográficas
  • Alexandersson, P., Féray, V.: Shifted symmetric functions and multirectangular coordinates of Young diagrams. J. Algebra 483, 262–305 (2017)
  • Bachmann, H.: The algebra of bi-brackets and regularized multiple Eisenstein series. J. Number Theory 200, 260–294 (2019)
  • Bachmann, H.: Multiple zeta values and modular forms. Lecture notes (Ver. 5.4). Nagoya University (2020). Available online at https://henrikbachmann.com/mzv2020.html
  • Bachmann, H., van Ittersum, J.-W.: Formal multiple Eisenstein series and their derivations, with an appendix by N. Matthes (in preparation)
  • Bachmann, H., Kühn, U.: The algebra of generating functions for multiple divisor sums and applications to multiple zeta values. Ramanujan...
  • Bachmann, H., Kühn, U.: A dimension conjecture for q-analogues of multiple zeta values, Periods in quantum field theory and arithmetic, Springer...
  • Bachmann, H., Kühn, U., Matthes, N.: Realizations of the formal double Eisenstein space, preprint (2021). arXiv:2109.04267
  • Bloch, S., Okounkov, A.: The character of the infinite wedge representation. Adv. Math. 149(1), 1–60 (2000)
  • Borwein, J.M., Bradley, D.M., Broadhurst, D.J., Lisonek, P.: Special values of multidimensional polylogarithms. Trans. Am. Math. Soc. 353,...
  • Borwein, J.M., Bradley, D.M., Broadhurst, D.J., Lisonek, P.: Combinatorial aspects of multiple zeta values. Electron. J. Combin. 5, 12 (1998)
  • Bradley, D.M.: Multiple q-zeta values. J. Algebra 283, 752–798 (2005)
  • Brindle, B.: Dualities of q-analogues of multiple zeta values. Master thesis, Universität Hamburg (2021). https://sites.google.com/view/benjamin-brindle/research
  • Brindle, B.: A unified approach to qMZVs, preprint (2021). arXiv:2111.00051
  • Chen, D., Möller, M., Zagier, D.: Quasimodularity and large genus limits of Siegel–Veech constants. J. Am. Math. Soc. 31(4), 1059–1163 (2018)
  • Dijkgraaf, R.: Mirror symmetry and elliptic curves, In The moduli space of curves (Texel Island, 1994), pp. 149–163. Progr. Math., 129, Birkhäuser...
  • Delecroix, V., Goujard, E., Zograf, P.: Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes, with an appendix by P....
  • Hoffman, M.E.: The algebra of multiple harmonic series. J. Algebra 194, 477–495 (1997)
  • Hoffman, M.E.: Quasi-shuffle products. J. Algebraic Combin. 11, 49–68 (2000)
  • Hoffman, M.E., Ihara, K.: Quasi-shuffle products revisited. J. Algebra 481, 293–326 (2017)
  • Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142, 307–338 (2006)
  • Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms. In: The moduli space of curves (Texel Island, 1994) (pp....
  • Matsumoto, K.: On analytic continuation of various multiple zeta-functions. Number theory for the millennium, II (Urbana, 2000), A. K. Peters,...
  • Okuda, J., Takeyama, Y.: On relations for the multiple q-zeta values. Ramanujan J. 14, 379–387 (2007)
  • Ohno, Y., Zagier, D.: Multiple zeta values of fixed weight, depth, and height. Indag. Math. 12(4), 483–487 (2001)
  • Pupyrev, Y.: Linear and algebraic independence of q-zeta values. Math. Notes 78(4), 563–568 (2005). Translated from Matematicheskie Zametki,...
  • Schlesinger, K.: Some remarks on q-deformed multiple polylogarithms, preprint (2011). arXiv /0111022
  • Singer, J.: q-Analogues of multiple zeta values and their application in renormalization, Dissertation, Erlangen-Nürnberg University (2017)
  • van Ittersum, J.W.M.: A symmetric Bloch–Okounkov theorem. Res. Math. Sci. 8(19), 42 (2021)
  • van Ittersum, J.W.M.: The Bloch–Okounkov theorem for congruence subgroups and Taylor coefficients of quasi-Jacobi forms. Res. Math. Sci. 10(5),...
  • Zagier, D.: Partitions, quasimodular forms, and the Bloch–Okounkov theorem. Ramanujan J. 41(1–3), 345–368 (2016)
  • Zagier, D.: The Mellin transform and other useful analytic techniques, Appendix to E. Zeidler, Quantum field theory I: Basics in mathematics...
  • Zhao, J.: Uniform approach to double shuffle and duality relations of various q-analogs of multiple zeta values via Rota-Baxter algebras....
  • Zudilin, V.V.: Algebraic relations for multiple zeta values. Uspekhi Mat. Nauk 58, 1(349), 3–32 (2013)
  • Zudilin, V.V.: Multiple q-zeta brackets. Mathematics 3, 119–130 (2015)