References
[As 70] ASAI, T.: On a certain function analogous to log|η(z)|.Nagoya Math. J. 40, (1970) 193–211.
[Ba 81] BARNER, K.: On Weil's explicit formula.J. reine angew. Math. 323, 139–152 (1981).
[Bo 51] BOCHNER, B.: Some properties of modular relations.Ann. Math. 53, (1951) 332–363.
[Ch 84] CHAVEL, I.:Eigenvalues in Riemannian Geometry. New York: Academic Press (1984).
[Cr 19] CRAMÉR, H.: Studien über die Nullstellen der Riemannschen Zetafunktion.Math. Z. 4, (1919) 104–130.
[Da 89] DAVIES, E. B.:Heat Kernels and Spectral Theory. Cambridge: Cambridge University Press (1989).
[Do 35] DOETSCH, G.: Summatorische Eigenschaften der Besselschen Funktionen und andere Funktionalrelationen, die mit der linearen Transformationsformel des Thetafunktionals äquivalent sind.Comp. Math. 1 (1934) 85–97.
[DG 75] DUISTERMAAT, J., and GUILLEMIN, V.: The spectrum of positive elliptic operators and periodic bicharacteristics.Invent. Math. 29, (1975) 39–79.
[Er 36a] ERDELYI, A.: Über eine Methode zur Gewinnung von Funktionalbeziehungen zwischen konfluenten hypergeometrischen Funktionen.Monat. f. Math. u. Phys. 45, (1936) 31–52.
[Er 36b] ERDELYI, A.: Funktionalrelationen mit konfluenten hypergeometrischen Funktionen, Erste Mitteilung: Additions- und Multiplikationstheoreme.Math. Zeitschr. 42, (1936) 125–143.
[Er 37a] ERDELYI, A.: Über gewisse Funktionalbeziehungen.Monat. f. Math. u. Phys. 45, (1937) 251–279.
[Ga 68] GANGOLLI, R.: Asymptotic behavior of spectra of compact quotients of certain symmetric spaces.Acta Math. 121 (1968) 151–192.
[Ga 77] GANGOLLI, R.: Zeta functions of Selberg's type for compact space forms of symmetric space of rank one.Ill. Math. J. 21, (1977) 1–42.
[GW 80] GANGOLLI, R., and WARNER, G.: Zeta functions of Selberg's type for some noncompact quotients of symmetric spaces of rank one.Nagoya Math. J. 78, (1980) 1–44.
[GR 65] GRADSHTEYN, I. S., and RYZHIK, I. M.:Tables of Integrals, Series, and Products. New York: Academic Press (1965).
[He 83] HEJHAL, D. A.: The Selberg Trace Formula forPSL (2,R), volume 2. Lecture Notes in Mathematics1001 Berlin-Heidelberg: Springer-Verlag (1983).
[Hel 64] HELGASON, S.: Fundamental solutions of invariant differential operators on symmetric spaces.Am. J. Math. 86, (1964) 565–601.
[Hel 78] HELGASON, S.:Differential Geometry, Lie Groups and Symmetric Spaces. New York: Academic Press (1978).
[Hel 84] HELGASON, S.:Groups and Geometric Analysis. New York: Academic Press (1978).
[Hel 94] HELGASON, S.:Geometric analysis on symmetric spaces. Math. Surveys and Monographs, AMS, 1994.
[Hu 84] HUXLEY, M. N.: “Scattering matrices for congruence subgroups,” in Modular Forms, R. A. Rankin ed., John Wiley and Sons: New York (1984) 157–196.
[JoL 93a] JORGENSON, J., and LANG, S.: Complex analytic properties of regularized products and series. Lecture Notes in Mathematics1564 Berlin-Heidelberg: Springer-Verlag (1993), 1–88.
[JoL 93b] JORGENSON, J., and LANG, S.: A Parseval formula for functions with a singular asymptotic expansion at the origin. Lecture Notes in Mathematics1564 Berlin-Heidelberg: Springer-Verlag (1993), 89–122.
[JoL 93c] JORGENSON, J., and LANG, S.: On Cramér's theorem for general Euler products with functional equation.Math. Ann. 297, (1993), 383–416.
[JoL 94a] JORGENSON, J., and LANG, S.: Artin formalism and heat kernels.J. reine angew. Math. 447, (1994) 165–200.
[JoL 94b] JORGENSON, J., and LANG, S.: Explicit formulas for regularized products and series. Lecture Notes in Mathematics1593 Berlin-Heidelberg: Springer-Verlag (1994), 1–134.
[Ko 35] KOBER, H.: Transformationalformeln gewisser Besselscher Reihen, Beziehungen zu Zeta-Funktionen.Math. Zeitschr. 39, (1935) 609–624.
[Kub 73] KUBOTA, T.:Elementary theory of Eisenstein series, New York: John Wiley and Sons (1973).
[La 70] LANG, S.:Algebraic Number Theory, Menlo Park, CA.: Addison-Wesley (1970); Graduate Texts in Mathematics110, New York: Springer-Verlag (1986); third edition, Springer-Verlag (1994).
[La 87] LANG, S.:Elliptic Functions, second edition. Graduate Texts in Mathematics112, New York: Springer-Verlag (1987).
[La 93a] LANG, S.:Complex Analysis, Graduate Texts in Mathematics103, New York: Springer-Verlag (1985), Third Edition (1993).
[La 93b] LANG, S.:Real and Functional Analysis, 3rd Edition, New York: Springer-Verlag (1993).
[La 95] LANG, S.:Differential and Riemannian Manifolds, New York: Springer-Verlag (1993).
[Lo 32] LOWRY, H.: Operational calculus II, The values of certain integrals and the relationships between various polynomials and series obtained by operational methods.Phil. Mag. (7)13 (1932) 1144–1163.
[Mil 78] MILLSON, J.: Closed geodesics and the η-invariant.Ann. Math. 108 (1978), 1–39.
[Sel 56] SELBERG, A.: Harmonic Analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series,J. Indian Math. Soc. B 20 (1956) 47–87. (Collected papers volume I, Berlin-Heidelberg: Springer-Verlag (1989) 423–463).
[Sel 90] SELBERG, A.: Remarks on the distribution of poles of Eisenstein series.Israel Mathematical Conference Proceedings 3 (1990) 251–278 (Collected papers volume II, Berlin-Heidelberg: Springer-Verlag (1991) 16–45.)
[Wa 44] WATSON, G. N.:A Treatise on the Theory of Bessel Functions, 2nd edition. Cambridge University Press: Cambridge (1944).
[We 52] WEIL, A.: Sur les “formules explicites” de la théorie des nombres premiers,Comm. Lund (vol. dédié à Marcel Riesz), 252–265 (1952).
[We 72] WEIL, A.: Sur les formules explicites de la théorie des nombres,Izv. Mat. Nauk (Ser. Mat.) 36, 3–18 (1972).
[Wo 67/72] WOLF, J.:Spaces of Constant Curvature, Berkeley (1967), second edition (1972).
Author information
Authors and Affiliations
Additional information
The first author acknowledges support from NSF grant DMS-93-07023 and from the Sloan Foundation. The second author thanks the Max-Planck-Institut for its yearly hospitality. We thank the referee for his corrections.
Rights and permissions
About this article
Cite this article
Jorgenson, J., Lang, S. Extension of analytic number theory and the theory of regularized harmonic series from Dirichlet series to Bessel series. Math. Ann. 306, 75–124 (1996). https://doi.org/10.1007/BF01445243
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01445243