Abstract
In this paper we will consider the functions E(z, ρ) obtained by setting the complex variable s in the Eisenstein series E(z, s) equal to a zero of the Riemann zeta-function and will show that these functions satisfy a number of remarkable relations. Although many of these relations are consequences of more or less well known identities, the interpretation given here seems to be new and of some interest. In particular, looking at the functions E(z, ρ) leads naturally to the definition of a certain representation of SL2(R) whose spectrum is related to the set of zeroes of the zeta-function.
Supported by the Sonderforschungsbereich „Theoretische Mathematik“ at the University of Bonn.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Bibliography
GELBART, S. and H. JACQUET,: A relation between automorphic representations of GL(2) and GL(3). Ann. Sc. Ec. Norm. Sup. 11 (1978) 471–542.
HECKE,E.: Über die Kroneckersche Grenzformel für reelle quadratische Körper und die Klassenzahl relativ-abelscher Körper, Verhandl. d. Naturforschenden Gesell. i. Basel 28, 363–372 (1917). Mathematische Werke, pp. 198–207. Vandenhoeck & Ruprecht, Göttingen 1970.
JACQUET, H. and J. SHALIKA,: A non-vanishing theorem for zeta functions of GL,. Invent. Math. 38 (1976) I - 16.
JACQUET, H. and D. ZAGIER,: Eisenstein series and and the Selberg trace formula II. In preparation.
RANKIN, R.: Contributions to the theory of Ramanujan’s functionand similar arithmetical functions. I. Proc. Cam. Phil. Soc. 35 (1939) 351–372.
SELBERG, A.: Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist. Arch. Math. Naturvid. 43 (1940) 47–50.
SHIMURA, G.: On the holomorphy of certain Dirichlet series. Proc. Lond. Math. Soc. 31 (1975), 79–98.
SHINTANI, T.: On zeta-functions associated with the vector space of quadratic forms J. Fac. Science Univ. Tokyo 22 (1975), 25–65.
SHINTANI, T.: On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J. 58 (1975), 83–126.
ZAGIER, D.: A Kronecker limit formula for real quadratic fields. Math. Ann. 213 (1975), 153–184.
ZAGIER, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In Modular Functions of One Variable VI, Lecture Notes in Mathematics No. 627, Springer, Berlin-Heidelberg—New York 1977, pp. 107–169.
ZAGIER, D.: Eisenstein series and the Selberg trace formula I. This volume, pp. 303–355.
ZAGIER, D.: The Rankin-Selberg method for automorphic functions which are not of rapid decay. In preparation.
Rights and permissions
Copyright information
© 1981 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zagier, D. (1981). Eisenstein Series and the Riemann Zeta-Function. In: Automorphic Forms, Representation Theory and Arithmetic. Tata Institute of Fundamental Research Studies in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-00734-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-662-00734-1_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-10697-5
Online ISBN: 978-3-662-00734-1
eBook Packages: Springer Book Archive