Analytic continuation of multiple zeta functions
HTML articles powered by AMS MathViewer
- by Jianqiang Zhao PDF
- Proc. Amer. Math. Soc. 128 (2000), 1275-1283 Request permission
Abstract:
In this paper we shall define the analytic continuation of the multiple (Euler-Riemann-Zagier) zeta functions of depth $d$: \[ \zeta (s_1,\dots ,s_d):= \sum _{0<n_1 < n_2<\cdots <n_d} \frac {1}{n_1^{s_1}n_2^{s_2}\cdots n_d^{s_d}},\] where $\operatorname {Re}(s_d)>1$ and $\sum _{j=1}^d\operatorname {Re}(s_j)>d$. We shall also study their behavior near the poles and pose some open problems concerning their zeros and functional equations at the end.References
- Tom M. Apostol and Thiennu H. Vu, Dirichlet series related to the Riemann zeta function, J. Number Theory 19 (1984), no. 1, 85–102. MR 751166, DOI 10.1016/0022-314X(84)90094-5
- David Borwein, Jonathan M. Borwein, and Roland Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 2, 277–294. MR 1335874, DOI 10.1017/S0013091500019088
- D.J. Broadhurst, Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams, preprint hep-th/9612012, available at http://xxx.lanl.gov/list/hep-th/9612.
- A. Beĭlinson and P. Deligne, Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 97–121 (French). MR 1265552
- L. Euler, Meditationes circa singvlare seriervm genus, Novi Comm. Acad. Sci. Petropol 20 (1775), pp. 140–186.
- I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
- Nobushige Kurokawa, Multiple zeta functions: an example, Zeta functions in geometry (Tokyo, 1990) Adv. Stud. Pure Math., vol. 21, Kinokuniya, Tokyo, 1992, pp. 219–226. MR 1210791, DOI 10.2969/aspm/02110219
- Tu Quoc Thang Le and Jun Murakami, Kontsevich’s integral for the Homfly polynomial and relations between values of multiple zeta functions, Topology Appl. 62 (1995), no. 2, 193–206. MR 1320252, DOI 10.1016/0166-8641(94)00054-7
- S. Lichtenbaum, Values of zeta functions at non-negative integers, in: Number theory, Noordwijkerhout 1983, Lecture Notes in Math. vol. 1068, Springer-Verlag, 1984, pp. 127–138.
- Don Zagier, Polylogarithms, Dedekind zeta functions and the algebraic $K$-theory of fields, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 391–430. MR 1085270, DOI 10.1007/978-1-4612-0457-2_{1}9
- Don Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992) Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 497–512. MR 1341859
Additional Information
- Jianqiang Zhao
- Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
- Address at time of publication: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
- Email: jzhao@math.brown.edu
- Received by editor(s): June 21, 1998
- Published electronically: August 5, 1999
- Communicated by: Dennis A. Hejhal
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1275-1283
- MSC (1991): Primary 11M99; Secondary 30D30, 30D10
- DOI: https://doi.org/10.1090/S0002-9939-99-05398-8
- MathSciNet review: 1670846