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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The distribution of smooth numbers in arithmetic progressions

Authors: Antal Balog and Carl Pomerance
Journal: Proc. Amer. Math. Soc. 115 (1992), 33-43
MSC: Primary 11N25; Secondary 11L05, 11N36
MathSciNet review: 1089401
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Abstract: We estimate the number of integers $ n$ up to $ x$ in the arithmetic progression $ a(\bmod q)$ with $ n$ free of prime factors exceeding $ y$. For a wide range of the variables $ x,y,q$, and $ a$ we show that this number is about $ x/(q{u^u})$, where $ u = \log x/\log y$.

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  • [de B] N. G. de Bruijn, The asymptotic behavior of a function occurring in the theory of primes, J. Indian Math. Soc. (NS) 15 (1951), 25-32. MR 0043838 (13:326f)
  • [CEP] E. R. Canfield, P. Erdös, and C. Pomerance, On a problem of Oppenheim concerning "Factorisatio Numerorum," J. Number Theory 17 (1983), 1-28. MR 712964 (85j:11012)
  • [F-T] E. Fouvry and G. Tenenbaum, Entiers sans grand facteur premier en progressions arithmétiques, Proc. London Math. Soc. (2) (to appear). MR 1127146 (93c:11074)
  • [Fl] J. B. Friedlander, Integers without large prime factors, Nederl. Akad. Wetensch. Proc. Ser A 76 (1973), 443-451. MR 0340201 (49:4957)
  • [F2] -, Integers without large prime factors. II, Acta Arith. 39 (1981), 53-57. MR 638742 (83b:10052)
  • [F-I] J. B. Friedlander and H. Iwaniec, On Bombieri's asymptotic sieve, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 719-756. MR 519891 (80j:10049)
  • [G] A. Granville, Integers, without large prime factors, in arithmetic progressions, Acta Math. (to appear). MR 1226529 (94f:11091)
  • [H-R] H. Halberstam and H. E. Richert, Sieve methods, Academic Press, London and New York, 1974. MR 0424730 (54:12689)
  • [Hi] A. Hildebrand, On the number of positive integers $ \leq x$ and free of prime factors $ > y$, J. Number Theory 22 (1986), 289-307. MR 831874 (87d:11066)
  • [Ho] C. Hooley, On the Brun-Titchmarsh theorem, J. Reine Angew. Math. 255 (1972), 60-79. MR 0304328 (46:3463)
  • [I] H. Iwaniec, On the Brun-Titchmarsh theorem, J. Math. Soc. Japan 34 (1982), 95-123. MR 639808 (83a:10082)
  • [L] M. Laborde, Nombres presque-premiers dans de petits intervalles, Sém. de Théorie des Nombres, Bordeaux, 1977-1978, exp. no. 15. MR 550275 (82b:10062)

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Article copyright: © Copyright 1992 American Mathematical Society

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