Absolute multiple sine functions. (English) Zbl 1425.11152
Summary: In this paper we formulate a unified theory of multiple sine functions by using a view point of absolute zeta functions and absolute automorphic forms.
MSC:
| 11M32 | Multiple Dirichlet series and zeta functions and multizeta values |
Keywords:
absolute multiple sine function; primitive multiple sine function; regularized multiple sine function; absolute zeta function; absolute automorphic formReferences:
| [1] | E. W. Barnes, On the theory of the multiple gamma function, Trans. Cambridge Philos. Soc. 19 (1904), 374-425. · JFM 35.0462.01 |
| [2] | L. Euler, De summis serierum reciprocarum, Commentarii academiae scientiarum Petropolitanae 7 (1740), 123-134. (Presented December 5, 1735), (Opera Omnia: Series 1, Volume 14, pp. 73-86).[E41] |
| [3] | O. Hölder, Ueber eine transcendente Function, Göttingen Nachrichten 1886 (1886), Nr. 16. 514-522. · JFM 18.0376.01 |
| [4] | N. Kurokawa, Multiple sine functions and Selberg zeta functions, Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), no. 3, 61-64. · Zbl 0738.11041 · doi:10.3792/pjaa.67.61 |
| [5] | N. Kurokawa, Gamma factors and Plancherel measures, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), no. 9, 256-260. · Zbl 0797.11053 · doi:10.3792/pjaa.68.256 |
| [6] | 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 |
| [7] | N. Kurokawa and S. Koyama, Multiple sine functions, Forum Math. 15 (2003), no. 6, 839-876. · Zbl 1065.11065 · doi:10.1515/form.2003.042 |
| [8] | N. Kurokawa and H. Ochiai, Dualities for absolute zeta functions and multiple gamma functions, Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 7, 75-79. · Zbl 1373.11063 · doi:10.3792/pjaa.89.75 |
| [9] | N. Kurokawa and H. Tanaka, Absolute zeta functions and the automorphy, Kodai Math. J. 40 (2017), no. 3, 584-614. · Zbl 1390.14067 · doi:10.2996/kmj/1509415235 |
| [10] | N. Kurokawa and H. Tanaka, Absolute zeta functions and absolute automorphic forms, J. Geom. Phys. 126 (2018), 168-180. · Zbl 1416.11137 · doi:10.1016/j.geomphys.2018.01.014 |
| [11] | N. Kurokawa and H. Tanaka, Limit formulas for multiple Hurwitz zeta functions, J. Number Theory 192 (2018), 348-355. · Zbl 1444.11187 · doi:10.1016/j.jnt.2018.04.021 |
| [12] | T. Shintani, On a Kronecker limit formula for real quadratic fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 167-199. · Zbl 0364.12012 |
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