The distribution of smooth numbers in arithmetic progressions

Balog, Antal; Pomerance, Carl

doi:10.1090/S0002-9939-1992-1089401-4

The distribution of smooth numbers in arithmetic progressions
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by Antal Balog and Carl Pomerance PDF

Proc. Amer. Math. Soc. 115 (1992), 33-43 Request permission

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 $u = \log x/\log y$.

References

N. G. de Bruijn, The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc. (N.S.) 15 (1951), 25–32. MR 43838
E. R. Canfield, Paul Erdős, and Carl Pomerance, On a problem of Oppenheim concerning “factorisatio numerorum”, J. Number Theory 17 (1983), no. 1, 1–28. MR 712964, DOI 10.1016/0022-314X(83)90002-1
É. Fouvry and G. Tenenbaum, Entiers sans grand facteur premier en progressions arithmetiques, Proc. London Math. Soc. (3) 63 (1991), no. 3, 449–494 (French). MR 1127146, DOI 10.1112/plms/s3-63.3.449
John B. Friedlander, Integers without large prime factors, Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math. 35 (1973), 443–451. MR 0340201
John B. Friedlander, Integers without large prime factors. II, Acta Arith. 39 (1981), no. 1, 53–57. MR 638742, DOI 10.4064/aa-39-1-53-57
John Friedlander and Henryk Iwaniec, On Bombieri’s asymptotic sieve, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 4, 719–756. MR 519891
Andrew Granville, Integers, without large prime factors, in arithmetic progressions. I, Acta Math. 170 (1993), no. 2, 255–273. MR 1226529, DOI 10.1007/BF02392787
H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR 0424730
Adolf Hildebrand, On the number of positive integers $\leq x$ and free of prime factors $>y$, J. Number Theory 22 (1986), no. 3, 289–307. MR 831874, DOI 10.1016/0022-314X(86)90013-2
Christopher Hooley, On the Brun-Titchmarsh theorem, J. Reine Angew. Math. 255 (1972), 60–79. MR 304328, DOI 10.1515/crll.1972.255.60
Henryk Iwaniec, On the Brun-Titchmarsh theorem, J. Math. Soc. Japan 34 (1982), no. 1, 95–123. MR 639808, DOI 10.2969/jmsj/03410095
Marc Laborde, Nombres presque-premiers dans de petits intervalles, Séminaire de Théorie des Nombres 1977–1978, CNRS, Talence, 1978, pp. Exp. No. 15, 17 (French). MR 550275