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Analytic continuation of multiple zeta functions


Author: Jianqiang Zhao
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
Published electronically: August 5, 1999
MathSciNet review: 1670846
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we shall define the analytic continuation of the multiple (Euler-Riemann-Zagier) zeta functions of depth $d$:

\begin{displaymath}\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}},\end{displaymath}

where $\re(s_d)>1$ and $\sum _{j=1}^d\re(s_j)>d$. We shall also study their behavior near the poles and pose some open problems concerning their zeros and functional equations at the end.


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  • 1. 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, https://doi.org/10.1016/0022-314X(84)90094-5
  • 2. 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, https://doi.org/10.1017/S0013091500019088
  • 3. 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.
  • 4. 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
  • 5. L. Euler, Meditationes circa singvlare seriervm genus, Novi Comm. Acad. Sci. Petropol 20 (1775), pp. 1-99.
  • 6. 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
    I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 2, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1968 [1977]. Spaces of fundamental and generalized functions; Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer. MR 0435832
    I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 3, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1967 [1977]. Theory of differential equations; Translated from the Russian by Meinhard E. Mayer. MR 0435833
    I. M. Gel′fand and N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Applications of harmonic analysis; Translated from the Russian by Amiel Feinstein. MR 0435834
    I. M. Gel′fand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1966 [1977]. Integral geometry and representation theory; Translated from the Russian by Eugene Saletan. MR 0435835
  • 7. 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
  • 8. 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, https://doi.org/10.1016/0166-8641(94)00054-7
  • 9. 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. CMP 16:17
  • 10. 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
  • 11. 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

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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

DOI: https://doi.org/10.1090/S0002-9939-99-05398-8
Keywords: Analytic continuation, multiple zeta function, generalized function
Received by editor(s): June 21, 1998
Published electronically: August 5, 1999
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 2000 American Mathematical Society