Advertisement

Archiv der Mathematik

, Volume 95, Issue 4, pp 363–372 | Cite as

Partial zeta functions

  • Yasufumi Hashimoto
Article

Abstract

In the present paper, we study analytic properties of the zeta functions defined by the Euler products over elements in subsets of the set of prime elements.

Mathematics Subject Classification (2000)

Primary: 11S40 Secondary: 11R42 11M36 

Keywords

Euler product Analytic continuation Natural boundary Dedekind zeta function Selberg zeta function Ihara zeta function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bauer P. J.: Zeros of Dirichlet L-series on the critical line. Acta Arith. 93, 37–52 (2000)MathSciNetGoogle Scholar
  2. 2.
    Bombieri E., Perelli A.: Distinct zeros of L-functions. Acta Arith. 83, 271–281 (1998)MATHMathSciNetGoogle Scholar
  3. 3.
    Conrey B.: More than two-fifth of the zeros of the Riemann zeta-function are on the critical line. J. Reine Angew. Math. 399, 1–26 (1989)MATHMathSciNetGoogle Scholar
  4. 4.
    Conrey B.: Zeros of derivatives of Riemann’s Xi-function on the critical line. II. J. Number Theory 17, 71–75 (1983)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Fujii A.: On the zeros of Dirichlet L-functions (V). Acta Arith. 28, 395–403 (1976)MATHGoogle Scholar
  6. 6.
    Hashimoto Y., Wakayama M.: Splitting density for lifting about discrete groups. Tohoku Math. J. 59, 527–545 (2007)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    D. Hejhal, The Selberg trace formula of \({PSL(2, \mathbb{R})}\) I, II, Lecture Notes in Math. 548, 1001 Springer-Verlag (1976, 1983).Google Scholar
  8. 8.
    M. N. Huxley, Scattering matrices for congruence subgroups, Modular forms (Ellis Horwood Ser. Math. Appl. 1984), 141–156.Google Scholar
  9. 9.
    Ihara Y.: On discrete subgroups of the two by two projective linear group over \({\mathfrak p}\)-adic fields. J. Math. Soc. Japan 18, 219–235 (1966)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    H. Iwaniec, Spectral Methods of Automorphic Forms, Graduate Studies in Mathematics, 53, 2nd ed., American Mathematical Society, (2002).Google Scholar
  11. 11.
    Koyama S.: Determinant expressions of Selberg zeta functions I. Trans. Amer. Math. Soc. 324, 149–168 (1991)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    N. Kurokawa On the meromorphy of Euler products I, II, Proc. London Math. Soc. (3) 53 (1986), 1–47(I) and 209–236(II).Google Scholar
  13. 13.
    Kurokawa N.: On certain Euler products. Acta Arith. 48, 49–52 (1987)MATHMathSciNetGoogle Scholar
  14. 14.
    N. Kurokawa, Analyticity of Dirichlet series over prime powers, Analytic number theory (Tokyo, 1988), 168–177, Lecture Notes in Math. 1434, Springer-Verlag, Berlin (1990).Google Scholar
  15. 15.
    Montgomery H.L.: Zeros of L-functions. Invent. Math. 8, 346–354 (1969)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Stark H.M., Terras A. A.: Zeta functions of finite graphs and coverings, II. Adv. Math. 154, 132–195 (2000)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    T. Sunada, L-functions in geometry and some applications, Curvature and topology of Riemannian manifolds (Katata, 1985), 266–284, Lecture Notes in Math. 1201, Springer-Verlag, Berlin (1986).Google Scholar
  18. 18.
    A. B. Venkov and P. G. Zograf, Analogues of Artin’s factorization formulas in the spectral theory of automorphic functions associated with induced representations of Fuchsian groups, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1150–1158, 1343 (Russian), Math. USSR-Izv. 21 (1983), 435–443 (English translation).Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Institute of Systems and Information Technologies/KyushuFukuokaJapan

Personalised recommendations