Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

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

  • [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,
  • [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,
  • [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,
  • [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.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11M06, 30B50, 30E25

Retrieve articles in all journals with MSC: 11M06, 30B50, 30E25

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society