Abstract
A multiple L-function and a multiple Hurwitz zeta function of EulerZagier type are introduced. Analytic continuation of them as complex functions of several variables is established by an application of the Euler-Maclaurin summation formula. Moreover location of singularities of such zeta functions is studied in detail.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. Akiyama, S. Egami, and Y. Tanigawa, An analytic continuation of multiple zeta functions and their values at non-positive integers, Acta Arith. 98 (2001), 107–116.
T. Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J. 153 (1999), 189–209.
F. V. Atkinson, The mean value of the Riemann zeta-function, Acta Math., 81 (1949), 353–376.
K. Dilcher, Zero of Bernoulli, generalized Bernoulli and Euler polynomials, Mem. Amer. Math. Soc., Number 386, 1988.
S. Egami, Introduction to multiple zeta function,Lecture Note at Niigata University (in Japanese).
K. Inkeri, The real roots of Bernoulli polynomials, Ann. Univ. Turku. Ser. A I 37 (1959), 20 pp.
M. Katsurada and K. Matsumoto, Asymptotic expansions of the mean values of Dirichlet L-functions. Math. Z., 208 (1991), 23–39.
Y. Motohashi, A note on the mean value of the zeta and L-functions. I, Proc. Japan Acad., Ser. A Math. Sci. 61 (1985), 222–224.
D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics,Vol. II, Birkhäuser, 1994, pp.497512.
D. Zagier, Periods of modular forms, traces of Hecke operators, and multiple zeta values, Research into automorphic forms and L functions (in Japanese) (Kyoto, 1992), Szirikaisekikenkyzisho Kbkyuroku, 843 (1993), 162–170.
J. Zhao, Analytic continuation of multiple zeta functions, Proc. Amer. Math. Soc., 128 (2000), 1275–1283.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Akiyama, S., Ishikawa, H. (2002). On Analytic Continuation of Multiple L-Functions and Related Zeta-Functions. In: Jia, C., Matsumoto, K. (eds) Analytic Number Theory. Developments in Mathematics, vol 6. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3621-2_1
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3621-2_1
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-5214-1
Online ISBN: 978-1-4757-3621-2
eBook Packages: Springer Book Archive