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Determinant expression of Selberg zeta functions. I


Author: Shin-ya Koyama
Journal: Trans. Amer. Math. Soc. 324 (1991), 149-168
MSC: Primary 11F72; Secondary 58G26
DOI: https://doi.org/10.1090/S0002-9947-1991-1041049-7
MathSciNet review: 1041049
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Abstract: We show that for $ {\text{PSL}}(2,{\mathbf{R}})$ and its congruence subgroup, the Selberg zeta function with its gamma factors is expressed as the determinant of the Laplacian. All the gamma factors are calculated explicitly. We also give an explicit computation to the contribution of the continuous spectrum to the determinant of the Laplacian.


References [Enhancements On Off] (What's this?)

  • [1] K. Doi and T. Miyake, Automorphic forms and number theory, Kinokuni-ya, 1976. (Japanese)
  • [2] J. Fischer, An approach to the Selberg trace formula via the Selberg zeta function, Lecture Notes in Math., vol. 1253, Springer, 1987. MR 892317 (88f:11053)
  • [3] M. N. Huxley, Scattering matrices for congruence subgroups, in Modular Forms (R. A. Rankin, ed.), Ellis Horwood, 1984. MR 803366 (87e:11072)
  • [4] H. von Kinkelin, Ueber eine mit der Gammafunktion verwandte Transcendente und deren Anwendung auf die Integralrechnung, J. Reine Angew. Math. 57 (1860), 122-138.
  • [5] N. Kurokawa, Parabolic components of zeta functions, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), 21-24. MR 953756 (89m:11052)
  • [6] P. Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), 113-120. MR 885573 (89e:58116)
  • [7] A. Selberg, Harmonic analysis and discontinuous groups on weakly symmetric Riemannian surfaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87. MR 0088511 (19:531g)
  • [8] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, Princeton, N.J., 1971. MR 0314766 (47:3318)
  • [9] A. B. Venkov, Spectral theory of automorphic functions, Proc. Steklov Inst. Math. 153 (1982), 1-163. MR 692019 (85j:11060b)
  • [10] M.-F. Vignéras, L'équation fonctionnelle de la fonction zêta de Selberg du groupe modulaire $ {\text{PSL}}(2,{\mathbf{Z}})$, Astérisque 61 (1979), 235-249.
  • [11] A. Voros, Spectral functions, special functions, and the Selberg zeta function, Comm. Math. Phys. 110 (1987), 439-465. MR 891947 (89b:58173)
  • [12] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Univ. Press, 1927. MR 1424469 (97k:01072)
  • [13] S. Koyama, Determinant expression of Selberg zeta functions. II, Trans. Amer. Math. Soc. (to appear). MR 1141858 (93g:11053)

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DOI: https://doi.org/10.1090/S0002-9947-1991-1041049-7
Article copyright: © Copyright 1991 American Mathematical Society

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