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On Bombieri's asymptotic sieve


Author: Kevin Ford
Journal: Trans. Amer. Math. Soc. 357 (2005), 1663-1674
MSC (2000): Primary 11N35
Published electronically: October 7, 2004
MathSciNet review: 2115380
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Abstract: If a sequence $(a_n)$ of non-negative real numbers has ``best possible'' distribution in arithmetic progressions, Bombieri showed that one can deduce an asymptotic formula for the sum $\sum_{n\le x} a_n \Lambda_k(n)$ for $k\ge 2$. By constructing appropriate sequences, we show that any weakening of the well-distribution property is not sufficient to deduce the same conclusion.


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

  • 1. Enrico Bombieri, On twin almost primes, Acta Arith. 28 (1975/76), no. 2, 177–193. MR 0396435
  • 2. Enrico Bombieri, The asymptotic sieve, Rend. Accad. Naz. XL (5) 1/2 (1975/76), 243–269 (1977) (English, with Italian summary). MR 0491570
  • 3. V. Brun, Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare, Archiv for Math. og Naturvid. B 34 (1915), no. 8, 19 pp.
  • 4. V. Brun, Le crible d'Eratosthéne et le théorème de Goldbach, Skr. Norske Vid.-Akad. Kristiania I. 1920, no. 3, 36 pp.
  • 5. John Friedlander and Henryk Iwaniec, Bombieri’s sieve, Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) Progr. Math., vol. 138, Birkhäuser Boston, Boston, MA, 1996, pp. 411–430. MR 1399351
  • 6. John Friedlander and Henryk Iwaniec, Asymptotic sieve for primes, Ann. of Math. (2) 148 (1998), no. 3, 1041–1065. MR 1670069, 10.2307/121035
  • 7. John Friedlander and Henryk Iwaniec, The polynomial 𝑋²+𝑌⁴ captures its primes, Ann. of Math. (2) 148 (1998), no. 3, 945–1040. MR 1670065, 10.2307/121034
  • 8. George Greaves, Sieves in number theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 43, Springer-Verlag, Berlin, 2001. MR 1836967
  • 9. H. Halberstam and H.-E. Richert, Sieve methods, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974. London Mathematical Society Monographs, No. 4. MR 0424730
  • 10. Atle Selberg, The general sieve-method and its place in prime number theory, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 1, Amer. Math. Soc., Providence, R. I., 1952, pp. 286–292. MR 0044563

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Additional Information

Kevin Ford
Affiliation: Department of Mathematics, 1409 West Green Sreet, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03579-2
Received by editor(s): September 16, 2003
Received by editor(s) in revised form: December 1, 2003
Published electronically: October 7, 2004
Additional Notes: This research was supported by National Science Foundation grants DMS-0070618 and DMS-0301083.
Article copyright: © Copyright 2004 American Mathematical Society