The Ramanujan Journal

, Volume 9, Issue 1–2, pp 139–202

• Régis De La Bretèche
• Gérald Tenenbaum
Article

## Abstract

Let Ψ(x,y) (resp. Ψ m (x,y)) denote the number of integers not exceeding x that are y-friable, i.e. have no prime factor exceeding y (resp. and are coprime to m). Evaluating the ratio Ψ m (x/d,y)/Ψ(x,y) for 1≤slantdslantx, m≥slant 1, x≥slant y≥slant 2, turns out to be a crucial step for estimating arithmetic sums over friable integers. Here, it is crucial to obtain formulae with a very wide range of validity. In this paper, several uniform estimates are provided for the aforementioned ratio, which supersede all previously known results. Applications are given to averages of various arithmetic functions over friable integers which in turn improve corresponding results from the literature. The technique employed rests mainly on the saddle-point method, which is an efficient and specific tool for the required design.

## Keywords

friable numbers saddle point method arithmetic functions

# Propriétés statistiques des entiers friables

## References

1. 1.
K. Alladi, “The Turán-Kubilius inequality for integers without large prime factors,” J. reine angew. Math. 335 (1982), 180–196.Google Scholar
2. 2.
R. de la Bretèche et G. Tenenbaum, “Sur les lois locales de la répartition du k-ième diviseur d’un entier,” Proc. London Math. Soc. 84(3) (2002), 289–323.Google Scholar
3. 3.
R. de la Bretèche et G. Tenenbaum, “Entiers friables: inégalité de Turán-Kubilius et applications,” Invent. Math. 159 (2005), 531–588.
4. 4.
N.G. de Bruijn, “On the number of positive integers ≤ x and free of prime factors > y, I, II,” Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951), 50–60; 69 (1966), 239–247.Google Scholar
5. 5.
H.G. Diamond and H. Halberstam, “The combinatorial sieve,” in Number Theory, Proc. 4th Matsci. Conf. Ootacamund/India 1984, Lect. Notes Math. 1122 (1985), 63–73.Google Scholar
6. 6.
E. Fouvry et G. Tenenbaum, “Entiers sans grand facteur premier en progressions arithmétiques,” Proc. London Math. Soc. 63(3) (1991), 449–494.Google Scholar
7. 7.
E. Fouvry et G. Tenenbaum, “Répartition statistique des entiers sans grand facteur premier dans les progressions arithmétiques,” Proc. London Math. Soc. 72(3) (1996), 481–514.Google Scholar
8. 8.
J.B. Friedlander, Integers free from large and small primes, Proc. London Math. Soc. 33(3) (1976), 565–576.Google Scholar
9. 9.
J.B. Friedlander and A. Granville, “Smoothing smooth numbers,” in Theory and Applications of Numbers Without Large Prime Factors (R.C. Vaughan, éd.), Phil. Trans. R. Soc. London A 345(1676) (1993), 339–347.Google Scholar
10. 10.
A. Granville, Integers, without large prime factors, in arithmetic progressions. II, in Theory and Applications of Numbers Without Large Prime Factors (R.C. Vaughan, éd.), Phil. Trans. R. Soc. London A 345 (1676) (1993), 349–362.Google Scholar
11. 11.
P.X. Gallagher, “A large sieve density estimate near σ = 1,” Inventiones Math. 11 (1970), 329–339.
12. 12.
D. Hensley, “A property of the counting function of integers with no large prime factors,” J. Number Theory 22(1) (1986), 46–74.
13. 13.
A. Hildebrand, “Integers free of large prime factors in short intervals,” Quart. J. Math. Oxford 36(2) (1985), 57–69.Google Scholar
14. 14.
A. Hildebrand, “On the number of positive integers ≤ x and free of prime factors > y,” J. Number Theory 22 (1986), 265–290.
15. 15.
A. Hildebrand, “On the local behaviour of Ψ(x, y),” Trans. Amer. Math. Soc. 295 (1986), 729–751.Google Scholar
16. 16.
A. Hildebrand and G. Tenenbaum, “On integers free of large prime factors,” Trans. Amer. Math. Soc. 296 (1986), 265–290.Google Scholar
17. 17.
A. Hildebrand and G. Tenenbaum, “On a class of differential–difference equations arising in number theory,” Journal d’analyse mathématique 61 (1993), 145–179.Google Scholar
18. 18.
A. Ivić, “On some estimates involving the number of prime divisor of an integers,” Acta Arith. 29 (1987), 21–33.Google Scholar
19. 19.
A. Ivić and G. Tenenbaum, “Local densities over integers free of large prime factors,” Quart. J. Math. (Oxford) 37(2) (1986), 401–417.Google Scholar
20. 20.
H.L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, Conference board of the mathematical sciences 84, American Mathematical Society, 1994.Google Scholar
21. 21.
H.L. Montgomery and R.C. Vaughan, “Mean values of multiplicative functions,” Period. Math. Hungar. 43(1/2) (2001), 199–214.
22. 22.
A. Rényi, “A new proof of a theorem of Delange,” Publ. Math. Debrecen 12 (1965), 323–329.Google Scholar
23. 23.
É. Saias, “Entiers sans grand ni petit facteur premier. III,” Acta Arith. 71(4) (1995), 351–379.Google Scholar
24. 24.
E. Scourfield, “On some sums involving the largest prime divisor of n,” Acta Arith. 59(4) (1991), 339–363.Google Scholar
25. 25.
E. Scourfield, “Comparison of two dissimilar sums involving the largest prime factor of an integer,” in Analytic Number Theory, (B.C. Berndt et al. éds.), Proceedings of a Conference Held in Honor of Heini Halberstam (Allerton Park, IL, 1995), Vol. 2, Prog. Math. 139 (1996), 723–735, Birkhäuser Boston, Boston, MA.Google Scholar
26. 26.
E. Scourfield, “On some sums involving the largest prime divisor of n, II,” Acta Arith. 98(4) (2001), 313–343.Google Scholar
27. 27.
H. Smida, “Sur les puissances de convolution de la fonction de Dickman,” Acta Arith. 59(2) (1991), 123-143.Google Scholar
28. 28.
G. Tenenbaum, “La méthode du col en Théorie Analytique des Nombres,” Séminaire de Théorie des nombres, Paris 1985–86, Prog. Math. 75 (1988), 411–441.Google Scholar
29. 29.
G. Tenenbaum, “Cribler les entiers sans grand facteur premier,” in Theory and Applications of Numbers Without Large Prime Factors (R.C. Vaughan, éd.), Phil. Trans. R. Soc. London A 345 (1993), 377–384.Google Scholar
30. 30.
G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Cours spécialisés, no. 1, Société Mathématique de France (1995), xv + 457 pp.Google Scholar
31. 31.
G. Tenenbaum, “Crible d’ératosthène et modèle de Kubilius,” in Number Theory in Progress (K. Györy, H. Iwaniec, J. Urbanowicz, éds.), Proceedings of the Conference in Honor of Andrzej Schinzel, Zakopane, Poland 1997, Walter de Gruyter, Berlin, New York, 1999, pp. 1099–1129.Google Scholar
32. 32.
G. Tenenbaum et J. Wu, Exercices corrigés de théorie analytique et probabiliste des nombres, Cours spécialisés, no. 2, Société Mathématique de France 1996.Google Scholar
33. 33.
E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nde édition, révisée par D.R. Heath-Brown, The Clarendon Press, Oxford University Press, New York, 1986.Google Scholar
34. 34.
T.Z. Xuan, “Integers free with no large prime factors,” Acta Arith. 69(4) (1995), 303–327.Google Scholar