Transactions of the American Mathematical Society

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On differential equations associated with Euler product expressions


Author: Ian Knowles
Journal: Trans. Amer. Math. Soc. 289 (1985), 545-573
MSC: Primary 11M06; Secondary 30B50, 30E25
MathSciNet review: 784003
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Abstract: A method is given by which one may associate (uniquely) certain differential equations with analytic functions defined by certain Euler product expressions. Some of the consequences of this construction include results relating the location of the zeros of the analytic functions to asymptotic properties of the solutions of the differential equations, as well as a differential equation characterization of those Dirichlet series with multiplicative coefficients.


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  • [1] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • [2] M. S. P. Eastham, Theory of ordinary differential equations, Van Nostrand Reinhold, London, 1970.
  • [3] Melvin Faierman and Ian Knowles, On a mixed problem for a hyperbolic equation with a discontinuity in the principal coefficients, Proc. London Math. Soc. (3) 46 (1983), no. 1, 137–166. MR 684826, 10.1112/plms/s3-46.1.137
  • [4] Ian Knowles, Eigenvalue problems and the Riemann zeta function. II, Ordinary differential equations and operators (Dundee, 1982) Lecture Notes in Math., vol. 1032, Springer, Berlin, 1983, pp. 267–297. MR 742644, 10.1007/BFb0076802
  • [5] Peter D. Lax and Ralph S. Phillips, Scattering theory for automorphic functions, Princeton Univ. Press, Princeton, N.J., 1976. Annals of Mathematics Studies, No. 87. MR 0562288
  • [6] Joseph Lehner, Lectures on modular forms, National Bureau of Standards, Applied Mathematics Series, vol. 61, Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1969. MR 0264070
  • [7] C. H. Müntz, Über den Approximationssatz von Weierstrass, Schwarz's Festschrift, Berlin, 1914, pp. 303-312.
  • [8] Andrew Ogg, Modular forms and Dirichlet series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0256993
  • [9] R. Paley and N. Weiner, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publ., vol. 19, Amer. Math. Soc., Providence, R. I., 1934.
  • [10] C. Ryavec, The analytic continuation of Euler products with applications to asymptotic formulae, Illinois J. Math. 17 (1973), 608–618. MR 0321894
  • [11] L. Schwartz, Etude des sommes d'exponentielles, Hermann, Paris, 1949.
  • [12] Otto Szász, Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen, Math. Ann. 77 (1916), no. 4, 482–496 (German). MR 1511875, 10.1007/BF01456964
  • [13] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485
  • [14] -, The theory of functions, 2nd ed., Oxford Univ. Press, London, 1939.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0784003-X
Article copyright: © Copyright 1985 American Mathematical Society