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Dualities for absolute zeta functions and multiple gamma functions. (English) Zbl 1373.11063
Summary: We study absolute zeta functions from the view point of a canonical normalization. We introduce the absolute Hurwitz zeta function for the normalization. In particular, we show that the theory of multiple gamma and sine functions gives good normalizations in cases related to the Kurokawa tensor product. In these cases, the functional equation of the absolute zeta function turns out to be equivalent to the simplicity of the associated non-classical multiple sine function of negative degree.

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M32 Multiple Dirichlet series and zeta functions and multizeta values 11M35 Hurwitz and Lerch zeta functions
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##### References:
 [1] E. W. Barnes, On the theory of the multiple gamma functions. Trans. Cambridge Philos. Soc. 19 (1904) 374-425. · JFM 35.0462.01 [2] A. Connes and C. Consani, Schemes over $$\mathbf{F}_{1}$$ and zeta functions, Compos. Math. 146 (2010), no. 6, 1383-1415. · Zbl 1201.14001 [3] A. Connes and C. Consani, Characteristic 1, entropy and the absolute point, in Noncommutative geometry, arithmetic, and related topics , Johns Hopkins Univ. Press, Baltimore, MD, 2011, pp. 75-139. · Zbl 1273.11140 [4] A. Deitmar, Remarks on zeta functions and $$K$$-theory over $$\mathbf{F}_{1}$$, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 8, 141-146. · Zbl 1173.14004 [5] N. Kurokawa, Multiple zeta functions: an example, in Zeta functions in geometry (Tokyo, 1990) , 219-226, Adv. Stud. Pure Math., 21, Kinokuniya, Tokyo, 1992. · Zbl 0795.11037 [6] N. Kurokawa, Zeta functions over $$\mathbf{F}_{1}$$, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 10, 180-184. · Zbl 1141.11316 [7] N. Kurokawa and S. Koyama, Multiple sine functions, Forum Math. 15 (2003), no. 6, 839-876. · Zbl 1065.11065 [8] N. Kurokawa and H. Ochiai, Multiple gamma functions of negative order. (Preprint). · Zbl 1373.11063 [9] Y. Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque 228 (1995), 121-163. · Zbl 0840.14001 [10] C. Soulé, Les variétés sur le corps à un élément, Mosc. Math. J. 4 (2004), no. 1, 217-244. · Zbl 1103.14003
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