Abstract
The functional determinant of an eigenvalue sequence, as defined by zeta regularization, can be simply evaluated by quadratures. We apply this procedure to the Selberg trace formula for a compact Riemann surface to find a factorization of the Selberg zeta function into two functional determinants, respectively related to the Laplacian on the compact surface itself, and on the sphere. We also apply our formalism to various explicit eigenvalue sequences, reproducing in a simpler way classical results about the gamma function and the BarnesG-function. Concerning the latter, our method explains its connection to the Selberg zeta function and evaluates the related Glaisher-Kinkelin constantA.
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D'Hoker, E., Phong, D. H.: (a) Multiloop amplitudes for the bosonic Polyakov string. Nucl. Phys.B269, 205–234 (1986)
(b) On determinants of Laplacians on Riemann surfaces. Commun. Math. Phys.104, 537–545 (1986)
Ray, D., Singer, I. M.: Ann. Math.98, 154–177 (1973)
Donnelly, H.: Am. J. Math.101, 1365–1379 (1979)
Fried, D.: Invent. Math.84, 523–540 (1986)
Widder, D.: The Laplace Transform (Chap. V), Princeton. NJ: Princeton University Press 1946
Duistermaat, H., Guillemin, V. W.: Invent. Math.29, 39–79 (1975)
Voros, A.: in: The Riemann problem.... Chudnovsky, D., Chudnovsky, G. (eds.) Lecture Notes in Mathematics Vol.925. Berlin, Heidelberg, New York: Springer 1982
Voros, A.: The return of the quartic oscillator. The complex WKB method. Ann. Inst. H. Poincaré39A, 211–338 (1983) (especially Sects. 4, 10 and Appendices A, C, D)
Hille, E.: Analytic function theory, Vol. I, Chap. 8.7 and Vol.II, Chap. 14, Blaisdell 1962–1963
Gelfand, I. M., Shilov, G. E.: Generalized functions Vol.1. New York: Academic Press 1964
Seeley, R.: AMS Proc. Symp. Pure Math.10, 288–307 (1966)
Gelfand, I. M., Levitan, B. M.: Dokl. Akad. Nauk. SSSR88, 593–596 (1953), Dikii, L. A.: Usp. Math. Nauk13, 111–143 (1958) (Translations AMS Series 218, 81–115)
Barnes, E. W.: Q. J. Math.31, 264–314 (1900)
Whittaker, E. T., Watson, G. N.: A course of modern analysis, Cambridge: Cambridge University Press 1965
Erdelyi et al.: Higher transcendental functions Vol.1, Chap. 1 (Bateman Manuscript Project), New York: McGraw Hill 1953
Selberg, A.: J. Ind. Math. Soc.20, 47–87 (1956)
Hejhal, D. A.: Duke Math. J.43, 441–482 (1976)
Balazs, N. L., Voros, A.: Chaos on the Pseudosphere. Phys. Rep.143, 109–240 (1986)
Huber, H.: Math. Anal.138, 1–26 (1959)
Belavin, A., Knizhnik, V.: JETP91, 364–390 (1986); Manin, YU.: JETP Lett.43, 161–163 (1986)
Fried, D.: Invent. Math.84, 523–540 (1986)
Kinkelin,: J. Reine Angew. Math. (Crelle)57, 122–138 (1860), Glaisher, J. W. L.: Messenger of Math.6, 71–76 (1877) and24, 1–16 (1894)
Cartier, P.: Analyse numérique d'un problème de valeurs propres à haute précision (Application aux fonctions automorphes), IHES preprint (1978); Vigneras, M-F.: Astérique61, 235–249 (1979)
Widom, H.: Indian Univ. Math. J.21, 277–283 (1971)
Widom, H.: Am. J. Math.95, 333–383 (1973); McCoy, B., Wu, T. T.: The two-dimensional Ising model, Cambridge, MA; Harvard University Press 1973 (page 264 and Appendix B); Dyson, F. J.: Fredholm determinants and inverse scattering problems. Commun. Math. Phys.47, 171–183 (1976)
Hardy, G. H.: Divergent Series, Clarendon Press, Oxford 1949
Gradshteyn, I. S., Ryzhik, I. M.: Tables of integrals, series and products (Corrected and Enlarged Edition prepared by A. Jeffrey), New York: Academic Press 1980
Lenard, A.: Pacific J. Math.42, 137–145 (1972)
Olver, F. W. J.: Asymptotics and special functions (Chap. 8, Sects. 2.2 and 3.3). New York: Academic Press 1974
Vardi, I.: Determinants of Laplacians and multiple gamma functions. Stanford Math. preprint (Sept. 1986), submitted to SIAM J. Math. Anal.; Weisberger, W. I.: Normalization of the path integral measure and the coupling constants for basonic strings, Nucl. Phys. B (in press) (1987)
Elstrodt, J.: Jber. d. Dt. Math. Verein83, 45–77 (1981), Eq. (10.5)
Randol, B.: Trans. AMS201, 241–246 (1975)
Selberg, A.: Lectures 1953–1954 (unpublished; private communication of J. Elstrodt); Randol, B.: Trans. AMS233, 241–247 (1977); Elstrodt, J., Grunewald, F., Mennicke, J.: Elementary and analytic theory of numbers. Banach center publications17, 83–120 (1985) (Warsaw)
Fischer, J.: Dissertation, Univ. Münster 1985 (and “An Approach to the Selberg trace formula via the Selberg zeta function”, submitted to Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer)
Sarnak, P.: Determinants of Laplacians, Comm. Math. Phys. (in press) (1987)
Balazs, N. L., Schmit, C., Voros, A.: Spectral fluctuations and zeta functions. Saclay preprint PhT/86-156. J. Stat. Phys. (to appear) (M. Kac memorial issue)
Voros, A.: Phys. Lett.B180, 245–246 (1986)
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Communicated by S-T. Yau
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Voros, A. Spectral functions, special functions and the Selberg zeta function. Commun.Math. Phys. 110, 439–465 (1987). https://doi.org/10.1007/BF01212422
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DOI: https://doi.org/10.1007/BF01212422