Proceedings of the Japan Academy, Series A, Mathematical Sciences

Values of absolute tensor products

Nobushige Kurokawa

Full-text: Open access


We study values of absolute tensor products (multiple zeta functions) at integral arguments. We obtain a simple formula for the absolute value of the double sine function. We express values of the multiple gamma function related to the functional equation.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 10 (2005), 185-190.

First available in Project Euclid: 28 December 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$

Absolute tensor product multiple zeta function multiple sine function multiple gamma function


Kurokawa, Nobushige. Values of absolute tensor products. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 10, 185--190. doi:10.3792/pjaa.81.185.

Export citation


  • E. W. Barnes, On the theory of the multiple gamma function, Trans. Cambridge Philos. Soc., 19 (1904), 374–425.
  • N. Kurokawa, Multiple zeta functions: an example, in Zeta functions in geometry (Tokyo, 1990), 219–226, Adv. Stud. Pure Math., 21, Kinokuniya, Tokyo.
  • N. Kurokawa, Derivatives of multiple sine functions, Proc. Japan Acad., 80A (2004), no. 5, 65–69.
  • N. Kurokawa, Zeta functions over $\mathbf{F}_1$, Proc. Japan Acad., 81A (2005), no. 10, 180–184.
  • N. Kurokawa and S. Koyama, Multiple sine functions, Forum Math. 15 (2003), no. 6, 839–876.
  • S. Koyama and N. Kurokawa, Kummer's formula for multiple gamma functions, J. Ramanujan Math. Soc. 18 (2003), no. 1, 87–107.
  • S. Koyama and N. Kurokawa, Multiple zeta functions: the double sine function and the signed double Poisson summation formula, Compos. Math. 140 (2004), no. 5, 1176–1190.
  • S. Koyama and N. Kurokawa, Multiple Euler Products, Proceedings of the St. Petersburg Math. Soc. 11 (2005) 123–166. (In Russian; The English version to be published from the American Math. Soc.).
  • S. Koyama and N. Kurokawa, Values of multiple zeta functions. (In preparation).
  • N. Kurokawa, H. Ochiai and M. Wakayama, Absolute derivations and zeta functions, Doc. Math. 2003, Extra Vol., 565–584 (electronic).
  • S. Lichtenbaum, Values of zeta-functions, étale cohomology, and algebraic $K$-theory, in Algebraic $K$-theory, II: “Classicalalgebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), 489–501. Lecture Notes in Math., 342, Springer, Berlin.
  • Yu. I. Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque No. 228 (1995), 4, 121–163.
  • Yu. I. Manin, The notion of dimension in geometry and algebra, (2005). (Preprint). math.AG/0502016.
  • D. Quillen, On the cohomology and $K$-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (1972), 552–586.
  • C. Soulé, Les variétés sur le corps à un élément, Mosc. Math. J. 4 (2004), no. 1, 217–244, 312.