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Investigations of Selberg Zeta Functions Under Character Deformations

  • Markus Szymon Fraczek
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2139)

Abstract

In this chapter we present the numerical results we obtained using our computer program package Morpheus for the transfer operators \(\mathcal{L}_{\beta,\varepsilon,\chi }^{\left (n\right )}\) in Sect.  7.6 and the Selberg zeta function \(Z^{\left (n\right )}(\beta,\chi )\) for \(\varGamma _{0}\left (n\right )\) and character χ given by
$$\displaystyle{Z^{\left (n\right )}(\beta,\chi ) =\det \left (1 -\tilde{\mathcal{L}}_{\beta,\chi }^{\left (n\right )}\right ) =\det \left (1 -\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )}\mathcal{L}_{\beta,-1,\chi }^{\left (n\right )}\right ) =\det \left (1 -\mathcal{L}_{\beta,-1,\chi }^{\left (n\right )}\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )}\right ).}$$

Keywords

Real Line Wave Form Transfer Operator Critical Line Riemann Zeta Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Supplementary material

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References

  1. 6.
    Avelin, H.: Deformation of Γ 0(5)-cusp forms. Math. Comput. 76, 361–384 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 22.
    Bruggeman, R., Fraczek, M., Mayer, D.: Perturbation of zeros of the Selberg zeta-function for Γ 0(4). Exp. Math. 22 (3), 217–252 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 30.
    Chang, C.H., Mayer, D.: An extension of the thermodynamic formalism approach to Selberg’s zeta function for general modular groups. In: Fiedler, B. (eds.) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp. 523–562. Springer, Berlin/New York (2001)CrossRefGoogle Scholar
  4. 59.
    Hejhal, D.: The Selberg Trace Formula for \(\mathrm{PSL}\!\left (2, \mathbb{R}\right )\), Volume 2. Lecture Notes in Mathematics, vol. 1001. Springer, Berlin/Heidelberg (1983)Google Scholar
  5. 73.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin/New York (1980)zbMATHGoogle Scholar
  6. 101.
    Petridis, Y., Risager, M.: Dissolving cusp forms: higher order Fermi’s golden rules. Mathematika 59, 269–301 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 103.
    Phillips, R., Sarnak, P.: The spectrum of fermat curves. Geom. Funct. Anal. 1, 80–146 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 104.
    Phillips, R., Sarnak, P.: Cusp forms for character varieties. Geom. Funct. Anal. 4, 93–118 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 119.
    Selberg, A.: Remarks on the distribution of poles of Eisenstein series. In: Festschrift in Honor of I.I. Piatetski-Shapiro, vol. 2, pp. 251–278 (1990) (Also in Collected Papers, vol. 2, pp. 15–45. Springer, Springer, Berlin/Heidelberg (1991))Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Markus Szymon Fraczek
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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