Illinois Journal of Mathematics

Generalization of the Wiener–Ikehara theorem

Gregory Debruyne and Jasson Vindas

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Abstract

We study the Wiener–Ikehara theorem under the so-called log-linearly slowly decreasing condition. Moreover, we clarify the connection between two different hypotheses on the Laplace transform occurring in exact forms of the Wiener–Ikehara theorem, that is, in “if and only if” versions of this theorem.

Article information

Source
Illinois J. Math., Volume 60, Number 2 (2016), 613-624.

Dates
Received: 29 November 2016
Revised: 30 January 2017
First available in Project Euclid: 11 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1499760025

Digital Object Identifier
doi:10.1215/ijm/1499760025

Mathematical Reviews number (MathSciNet)
MR3680551

Zentralblatt MATH identifier
1382.40018

Subjects
Primary: 11M45: Tauberian theorems [See also 40E05] 40E05: Tauberian theorems, general

Citation

Debruyne, Gregory; Vindas, Jasson. Generalization of the Wiener–Ikehara theorem. Illinois J. Math. 60 (2016), no. 2, 613--624. doi:10.1215/ijm/1499760025. https://projecteuclid.org/euclid.ijm/1499760025


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References

  • J. Aramaki, An extension of the Ikehara Tauberian theorem and its application, Acta Math. Hungar. 71 (1996), 297–326.
  • J. J. Benedetto, Spectral synthesis, Academic Press, Inc., New York–London, 1975.
  • H. Bremermann, Distributions, complex variables and Fourier transforms, Addison-Wesley, Reading, MA, 1965.
  • G. Debruyne and J. Vindas, On PNT equivalences for Beurling numbers, to appear in Monatsh. Math. DOI:\doiurl10.1007/s00605-016-0979-9.
  • G. Debruyne and J. Vindas, Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior, to appear in J. Anal. Math.; available at \arxivurlarXiv:1604.05069.
  • H. Delange, Généralisation du théorème de Ikehara, Ann. Sci. Éc. Norm. Supér. (4) 71 (1954), 213–242.
  • H. G. Diamond and W.-B. Zhang, Chebyshev bounds for Beurling numbers, Acta Arith. 160 (2013), 143–157.
  • H. G. Diamond and W.-B. Zhang, Beurling generalized numbers, Mathematical Surveys and Monographs Series, American Mathematical Society, Providence, RI, 2016.
  • S. W. Graham and J. D. Vaaler, A class of extremal functions for the Fourier transform, Trans. Amer. Math. Soc. 265 (1981), 283–302.
  • S. Ikehara, An extension of Landau's theorem in the analytic theory of numbers, J. Math. and Phys. M.I.T. 10 (1931), 1–12.
  • J.-P. Kahane and R. Salem, Ensembles parfaits et séries trigonométriques, 2nd ed., Hermann, Paris, 1994.
  • J. Korevaar, Tauberian theory. A century of developments, Grundlehren der Mathematischen Wissenschaften, vol. 329, Springer-Verlag, Berlin, 2004.
  • J. Korevaar, Distributional Wiener–Ikehara theorem and twin primes, Indag. Math. (N.S.) 16 (2005), 37–49.
  • S. Pilipović, B. Stanković and J. Vindas, Asymptotic behavior of generalized functions, Series on Analysis, Applications and Computation, vol. 5, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.
  • S. G. Révész and A. de Roton, Generalization of the effective Wiener–Ikehara theorem, Int. J. Number Theory 9 (2013), 2091–2128.
  • W. Rudin, Lectures on the edge-of-the-wedge theorem, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, vol. 6, AMS, Providence, RI, 1971.
  • J.-C. Schlage-Puchta and J. Vindas, The prime number theorem for Beurling's generalized numbers. New cases, Acta Arith. 153 (2012), 299–324.
  • V. S. Vladimirov, Methods of the theory of generalized functions, analytical methods and special functions, vol. 6, Taylor & Francis, London, 2002.
  • N. Wiener, The Fourier integral and certain of its applications, Cambridge University Press, Cambridge, MA, 1988. Reprint of the 1933 edition.
  • W.-B. Zhang, Wiener–Ikehara theorems and the Beurling generalized primes, Monatsh. Math. 174 (2014), 627–652.