Skip to main content Accessibility help
×
Home

ON WITTEN MULTIPLE ZETA-FUNCTIONS ASSOCIATED WITH SEMI-SIMPLE LIE ALGEBRAS V

  • YASUSHI KOMORI (a1), KOHJI MATSUMOTO (a2) and HIROFUMI TSUMURA (a3)

Abstract

We study the values of the zeta-function of the root system of type G2 at positive integer points. In our previous work we considered the case when all integers are even, but in the present paper we prove several theorems which include the situation when some of the integers are odd. The underlying reason why we may treat such cases, including odd integers, is also discussed.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      ON WITTEN MULTIPLE ZETA-FUNCTIONS ASSOCIATED WITH SEMI-SIMPLE LIE ALGEBRAS V
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      ON WITTEN MULTIPLE ZETA-FUNCTIONS ASSOCIATED WITH SEMI-SIMPLE LIE ALGEBRAS V
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      ON WITTEN MULTIPLE ZETA-FUNCTIONS ASSOCIATED WITH SEMI-SIMPLE LIE ALGEBRAS V
      Available formats
      ×

Copyright

References

Hide All
1.Apostol, T. M., Introduction to analytic number theory (Springer, New York, NY, 1976).
2.Bourbaki, N., Groupes et algèbres de Lie, chapitres 4, 5 et 6 (Hermann, Paris, France, 1968).
3.Humphreys, J. E., Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics 9 (Springer-Verlag, New York, NY, 1972).
4.Humphreys, J. E., Reflection groups and coxeter groups (Cambridge University Press, Cambridge, UK, 1990).
5.Komori, Y., Matsumoto, K. and Tsumura, H., Zeta-functions of root systems, in Proceedings of the conference on L-functions, Fukuoka, Japan, 2006) (Weng, L. and Kaneko, M., Editors) (World Science Publisher, Hackensack, NJ, 2007), 115140.
6.Komori, Y., Matsumoto, K. and Tsumura, H., Zeta and L-functions and Bernoulli polynomials of root systems, Proc. Japan Acad. Ser. A 84 (2008), 5762.
7.Komori, Y., Matsumoto, K. and Tsumura, H., On Witten multiple zeta-functions associated with semisimple Lie algebras II, J. Math. Soc. Japan 62 (2010), 355394.
8.Komori, Y., Matsumoto, K. and Tsumura, H., On multiple Bernoulli polynomials and multiple L-functions of root systems, Proc. London Math. Soc. 100 (2010), 303347.
9.Komori, Y., Matsumoto, K. and Tsumura, H., Functional relations for zeta-functions of root systems, in Number theory: Dreaming in dreams – proceedings of the 5th China-Japan seminar (Aoki, T., Kanemitsu, S. and Liu, J.-Y., Editors) (World Science Publisher, Hackensack, NJ, 2010), 135183.
10.Komori, Y., Matsumoto, K. and Tsumura, H., On Witten multiple zeta-functions associated with semisimple Lie algebras III, in Multiple Dirichlet series, L-functions and automorphic forms (Bump, D., Friedberg, S. and Goldfeld, D., Editors), Progress in Mathematics Series, vol. 300 (Birkhäuser/Springer, New York, NY, 2012) 223286.
11.Komori, Y., Matsumoto, K. and Tsumura, H., On Witten multiple zeta-functions associated with semisimple Lie algebras IV, Glasgow Math. J. 53 (2011), 185206.
12.Komori, Y., Matsumoto, K. and Tsumura, H., Zeta-functions of weight lattices of compact connected semisimple Lie groups, preprint, arXiv:math/1011.0323.
13.Matsumoto, K., Nakamura, T., Ochiai, H. and Tsumura, H., On value-relations, functional relations and singularities of Mordell-Tornheim and related triple zeta-functions, Acta Arith. 132 (2008), 99125.
14.Matsumoto, K., Nakamura, T. and Tsumura, H., Functional relations and special values of Mordell-Tornheim triple zeta and L-functions, Proc. Amer. Math. Soc. 136 (2008), 21352145.
15.Matsumoto, K. and Tsumura, H., On Witten multiple zeta-functions associated with semisimple Lie algebras I, Ann. Inst. Fourier 56 (2006), 14571504.
16.Nakamura, T., A functional relation for the Tornheim double zeta function, Acta Arith. 125 (2006), 257263.
17.Nakamura, T., Double Lerch series and their functional relations, Aequationes Math. 75 (2008), 251259.
18.Nakamura, T., Double Lerch value relations and functional relations for Witten zeta functions, Tokyo J. Math. 31 (2008), 551574.
19.Okamoto, T., Multiple zeta values related with the zeta-function of the root system of type A 2, B 2 and G 2, Comment. Math. Univ. St. Pauli 61 (2012), 927.
20.Onodera, K., Generalized log sine integrals and the Mordell–Tornheim zeta values, Trans. Amer. Math. Soc. 363 (2011), 14631485.
21.Tornheim, L., Harmonic double series, Amer. J. Math. 72 (1950), 303314.
22.Tsumura, H., On Witten's type of zeta values attached to SO(5), Arch. Math. (Basel) 82 (2004), 147152.
23.Witten, E., On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), 153209.
24.Zagier, D., Values of zeta functions and their applications, in First European Congress of Mathematics vol. II (Joseph, A.et al. Editors), Progress in Mathematics Series, vol. 120 (Birkhäuser, Basel, Switzerland, 1994), 497512.
25.Zagier, D., Introduction to multiple zeta values, Lectures at Kyushu University (1999, unpublished note).
26.Zhao, J., Multi-polylogs at twelfth roots of unity and special values of Witten multiple zeta function attached to the exceptional Lie algebra $\mathfrak{g}_2$, J. Algebra Appl. 9 (2010), 327337.

MSC classification

ON WITTEN MULTIPLE ZETA-FUNCTIONS ASSOCIATED WITH SEMI-SIMPLE LIE ALGEBRAS V

  • YASUSHI KOMORI (a1), KOHJI MATSUMOTO (a2) and HIROFUMI TSUMURA (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed