A $q$-analogue of multiple zeta values and its application to number theory (Various aspects of integrable systems)

[1] Bachmann, H., Takeyama, Y. and Tasaka, K., Cyclotomic analogues of finite multiple zeta

values, Compos. Math. 154 (2018), no. 12, 2701–2721.

[2] Bachmann, H., Takeyama, Y. and Tasaka, K., Finite and symmetric Mordell-Tornheim

multiple zeta values, J. Math. Soc. Japan 73 (2021), no. 4, 1129–1158.

[3] Bradley, D., Multiple q-zeta values, J. Algebra 283 (2005), no. 2, 752–798.

[4] Bradley, D. and Zhou, X., On Mordell-Tornheim sums and multiple zeta values, Ann. Sci.

Math. Qu´ebec 34 (2010), no. 1, 15–23.

[5] Deligne, P., Goncharov, A. B., Groupes fondamentaux motiviques de Tate mixte, Ann.

Sci. Ecole

Norm. Sup. (4) 38 (2005), no. 1, 1–56.

[6] Granville, A., A decomposition of Riemann’s zeta-function, Analytic number theory (Kyoto, 1996), 95–101, London Math. Soc. Lecture Note Ser., 247, Cambridge Univ. Press,

Cambridge, 1997.

[7] Goncharov, A.B., Periods and mixed motives, preprint, 2002.

[8] Hoffman, M., Quasi-symmetric functions and mod p multiple harmonic sums, Kyushu J.

Math. 69 (2015), no. 2, 345–366.

[9] Ihara, K., Kaneko, M. and Zagier, D., Derivation and double shuffle relations for multiple

zeta values, Compos. Math. 142 (2006), no. 2, 307–338.

[10] Jarossay, D., Double m´elange des multizˆetas finis et multizˆetas sym´etris´es, C. R. Math.

Acad. Sci. Paris 352 (2014), no. 10, 767–771.

[11] Kamano, K., Finite Mordell-Tornheim multiple zeta values, Funct. Approx. Comment.

Math. 54 (2016), 65–72.

[12] Kaneko, M., An introduction to classical and finite multiple zeta values, Publications

Math´ematiques de Besan¸con - Alg´ebre et Th´eorie des Nombres, no. 1 (2019), 103–129.

[13] Kaneko, M., Kurokawa, N. and Wakayama, M., A variation of Euler’s approach to values

of the Riemann zeta function, Kyushu J. Math., 57 (2003), no. 1, 175–192.

[14] Kaneko, M. and Zagier, D., Finite multiple zeta values, in preparation.

[15] Kawashima, G., A class of relations among multiple zeta values, J. Number Theory 129

(2009), no. 4, 755–788.

[16] Matsumoto, K., On Mordell-Tornheim and other multiple zeta-functions, in Proceedings

of the Session in Analytic Number Theory and Diophantine Equations, Bonner Math.

Schriften, 360, Univ. Bonn, Bonn, 2003, n.25, 17pp.

86

Yoshihiro Takeyama

[17] Mordell, L. J., On the evaluation of some multiple series. J. London Math. Soc. 33 (1958),

368–371.

[18] Ohno, Y., A generalization of the duality and sum formulas on the multiple zeta values,

J. Number Theory 74 (1999), no. 1, 39–43.

[19] Ohno, Y. and Zagier, D., Multiple zeta values of fixed weight, depth, and height, Indag.

Math. (N.S.) 12 (2001), no. 4, 483–487.

[20] Okuda, J. and Takeyama, Y., On relations for the multiple q-zeta values, Ramanujan J.

14 (2007), no. 3, 379–387.

[21] Oyama, K., Ohno-type relation for finite multiple zeta values, Kyushu J. Math. 72 (2018),

no. 2, 277–285.

[22] Seki, S., Connectors, in Various aspects of multiple zeta values, RIMS kˆ

okyˆ

uroku, 2160

(2020), 15–27.

[23] Seki, S. and Yamamoto, S., A new proof of the duality of multiple zeta values and its

generalizations, Int. J. Number Theory 15 (2019), no. 6, 1261–1265.

[24] Takeyama, Y., Quadratic relations for a q-analogue of multiple zeta values, Ramanujan J.

27 (2012), no. 1, 15–28.

[25] Takeyama, Y., Derivations on the algebra of multiple harmonic q-series and their applications, Ramanujan J. 52 (2020), no. 1, 41–65.

[26] Terasoma, T., Mixed Tate motives and multiple zeta values, Invent. Math. 149 (2002),

no. 2, 339–369.

[27] Tornheim, L., Harmonic double series, Amer. J. Math., 72 (1950), 303–314.

[28] Yamamoto, S., Multiple zeta-star values and multiple integrals, in Various Aspects of

Multiple Zeta Values, RIMS Kˆ

okyˆ

uroku Bessatsu, B68, Res. Inst. Math. Sci. (RIMS),

Kyoto, 2017, 3–14.

[29] Yasuda, S., Finite real multiple zeta values generate the whole space Z, Int. J. Number

Theory 12 (2016), 787–812.

[30] Zagier, D., Values of zeta functions and their applications, First European Congress of

Mathematics, Vol. II (Paris, 1992), 497–512, Progr. Math., 120, Birkh¨

auser, Basel, 1994.

[31] Zagier, D., Multiple zeta values, unpublished manuscript, 1995.

[32] Zhao, J., Multiple q-zeta functions and multiple q-polylogarithms, Ramanujan J., 14

(2007), no. 2, 189–221.

[33] Zhao, J., Multiple zeta functions, multiple polylogarithms and their special values, Series

on Number Theory and its Applications 12, World Scientific Publishing Co., 2016.

[34] Zhao, J., Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory

4 (2008), no. 1, 73–106.

...