Determinant expression of Selberg zeta functions. I
HTML articles powered by AMS MathViewer
- by Shin-ya Koyama
- Trans. Amer. Math. Soc. 324 (1991), 149-168
- DOI: https://doi.org/10.1090/S0002-9947-1991-1041049-7
- PDF | Request permission
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
- K. Doi and T. Miyake, Automorphic forms and number theory, Kinokuni-ya, 1976. (Japanese)
- Jürgen Fischer, An approach to the Selberg trace formula via the Selberg zeta-function, Lecture Notes in Mathematics, vol. 1253, Springer-Verlag, Berlin, 1987. MR 892317, DOI 10.1007/BFb0077696
- M. N. Huxley, Scattering matrices for congruence subgroups, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 141–156. MR 803366 H. von Kinkelin, Ueber eine mit der Gammafunktion verwandte Transcendente und deren Anwendung auf die Integralrechnung, J. Reine Angew. Math. 57 (1860), 122-138.
- Nobushige Kurokawa, Parabolic components of zeta functions, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), no. 1, 21–24. MR 953756
- Peter Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), no. 1, 113–120. MR 885573
- A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
- Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. MR 0314766
- A. B. Venkov, Spectral theory of automorphic functions, Trudy Mat. Inst. Steklov. 153 (1981), 172 (Russian). MR 665585 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.
- A. Voros, Spectral functions, special functions and the Selberg zeta function, Comm. Math. Phys. 110 (1987), no. 3, 439–465. MR 891947
- E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759
- Shin-ya Koyama, Determinant expression of Selberg zeta functions. II, Trans. Amer. Math. Soc. 329 (1992), no. 2, 755–772. MR 1141858, DOI 10.1090/S0002-9947-1992-1141858-0
Bibliographic Information
- © Copyright 1991 American Mathematical Society
- 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