Illinois Journal of Mathematics

Evaluation of Tornheim’s type of double series

Shin-ya Kadota, Takuya Okamoto, and Koji Tasaka

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We examine values of certain Tornheim’s type of double series with odd weight. As a result, an affirmative answer to a conjecture about the parity theorem for the zeta function of the root system of the exceptional Lie algebra $G_{2}$, proposed by Komori, Matsumoto and Tsumura, is given.

Article information

Illinois J. Math., Volume 61, Number 1-2 (2017), 171-186.

Received: 29 March 2017
Revised: 3 September 2017
First available in Project Euclid: 3 March 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Secondary: 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)


Kadota, Shin-ya; Okamoto, Takuya; Tasaka, Koji. Evaluation of Tornheim’s type of double series. Illinois J. Math. 61 (2017), no. 1-2, 171--186. doi:10.1215/ijm/1520046214.

Export citation


  • T. Arakawa, T. Ibukiyama and M. Kaneko, Bernoulli numbers and zeta functions, Springer Monographs in Mathematics, Springer, Tokyo, 2014.
  • J. G. Huard, K. S. Williams and N. Y. Zhang, On Tornheim's double series, Acta Arith. 75 (1996), no. 2, 105–117.
  • K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006), no. 2, 307–338.
  • Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semi-simple Lie algebras IV, Glasg. Math. J. 53 (2011), no. 1, 185–206.
  • Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zeta-function associated with semi-simple Lie algebras V, Glasg. Math. J. 57 (2015), no. 1, 107–130.
  • T. Nakamura, A functional relation for the Tornheim double zeta function, Acta Arith. 125 (2006), no. 3, 257–263.
  • T. Nakamura, A simple proof of the functional relation for the Lerch type Tornheim double zeta function, Tokyo J. Math. 35 (2012), no. 2, 333–337.
  • T. Okamoto, 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), no. 1, 9–27.
  • T. Okamoto, On alternating analogues of the Mordell–Tornheim triple zeta values, J. Ramanujan Math. Soc. 28 (2013), no. 2, 247–269.
  • E. Panzer, The parity theorem for multiple polylogarithms, J. Number Theory 172 (2017), 93–113.
  • M. V. Subbarao and R. Sitaramachandra, On some infinite series of L. J. Mordell and their analogues, Pacific J. Math. 119 (1985), no. 1, 245–255.
  • L. Tornheim, Harmonic double series, Amer. J. Math. 72 (1950), 303–314.
  • H. Tsumura, On alternating analogues of Tornheim's double series, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3633–3641.
  • H. Tsumura, Evaluation formulas for Tornheim's type of alternating double series, Math. Comp. 73 (2004), no. 245, 251–258.
  • H. Tsumura, Combinatorial relations for Euler–Zagier sums, Acta Arith. 111 (2004), no. 1, 27–42.
  • H. Tsumura, On Mordell-Tornheim zeta values, Proc. Amer. Math. Soc. 133 (2005), 2387–2393.
  • H. Tsumura, On alternating analogues of Tornheim's double series. II, Ramanujan J. 18 (2009), no. 1, 81–90.
  • J. Zhao, 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), no. 2, 327–337.
  • X. Zhou, T. Cai and D. M. Bradley, Signed $q$-analogs of Tornheim's double series, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2689–2698.