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The Wiener-Ikehara theorem by complex analysis

Author: Jaap Korevaar
Journal: Proc. Amer. Math. Soc. 134 (2006), 1107-1116
MSC (2000): Primary 40E05; Secondary 11M45, 11N05, 44A10
Published electronically: August 12, 2005
MathSciNet review: 2196045
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Abstract: The Tauberian theorem of Wiener and Ikehara provides the most direct way to the prime number theorem. Here it is shown how Newman's contour integration method can be adapted to establish the Wiener-Ikehara theorem. A simple special case suffices for the PNT. But what about the twin-prime problem?

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  • 1. Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, vol. 96, Birkhäuser Verlag, Basel, 2001. MR 1886588
  • 2. R.F. Arenstorf, There are infinitely many prime twins, Preprint, Vanderbilt University, Nashville, TN; 38 pages. Posted on the arXiv:math.NT/0405509, May 26, 2004; withdrawn June 9, 2004. Available at
  • 3. S. Bochner, Ein Satz von Landau und Ikehara, Math. Z. 37 (1933), 1-9.
  • 4. H.G. Diamond, Elementary methods in the study of the distribution of prime numbers, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 553-589. MR 0670132 (83m:10002)
  • 5. S.W. Graham, J.D. Vaaler, A class of extremal functions for the Fourier transform, Trans. Amer. Math. Soc. 265 (1981), 283-302. MR 0607121 (82i:42008)
  • 6. G.H. Hardy, J.E. Littlewood, Some problems of `partitio numerorum' III, On the expression of a number as a sum of primes, Acta Math. 44 (1922), 1-70.
  • 7. S. Ikehara, An extension of Landau's theorem in the analytic theory of numbers, J. Math. and Phys. 10 (1931), 1-12.
  • 8. A. E. Ingham, The distribution of prime numbers, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original; With a foreword by R. C. Vaughan. MR 1074573
  • 9. J. Korevaar, On Newman's quick way to the prime number theorem, Math. Intelligencer 4 (1982), 108-115. MR 0684025 (84b:10063)
  • 10. J. Korevaar, A century of complex Tauberian theory, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 4, 475–531. MR 1920279,
  • 11. Jacob Korevaar, Tauberian theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 329, Springer-Verlag, Berlin, 2004. A century of developments. MR 2073637
  • 12. J. Korevaar, Distributional Wiener-Ikehara theorem and twin primes, Indag. Math. (N.S.) 16 (2005), 37-49.
  • 13. Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen. 2 Bände, Chelsea Publishing Co., New York, 1953 (German). 2d ed; With an appendix by Paul T. Bateman. MR 0068565
  • 14. E. Landau, Über den Wienerschen neuen Weg zum Primzahlsatz, Sitz.-Ber. Preuss. Akad. Wiss., Phys.-Math. Klasse, 1932, 514-521.
  • 15. Władysław Narkiewicz, The development of prime number theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. From Euclid to Hardy and Littlewood. MR 1756780
  • 16. D.J. Newman, Simple analytic proof of the prime number theorem, Amer. Math. Monthly 87 (1980), 693-696. MR 0602825 (82h:10056)
  • 17. Gérald Tenenbaum and Michel Mendès France, The prime numbers and their distribution, Student Mathematical Library, vol. 6, American Mathematical Society, Providence, RI, 2000. Translated from the 1997 French original by Philip G. Spain. MR 1756233
  • 18. Norbert Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), no. 1, 1–100. MR 1503035,
  • 19. N. Wiener, The Fourier Integral and Certain of its Applications, Cambridge Univ. Press, Cambridge, 1933. (Reprinted 1988.) MR 0983891 (90b:42022)
  • 20. D. Zagier, Newman’s short proof of the prime number theorem, Amer. Math. Monthly 104 (1997), no. 8, 705–708. MR 1476753,

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Additional Information

Jaap Korevaar
Affiliation: KdV Institute of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, Netherlands

Keywords: Laplace transform, prime number theorem, Tauberian theory, Wiener--Ikehara theorem
Received by editor(s): April 20, 2004
Received by editor(s) in revised form: November 2, 2004
Published electronically: August 12, 2005
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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