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Entiers friables dans des progressions arithmétiques de grand module

  • R. DE LA BRETÈCHE (a1) and D. FIORILLI (a1)

We study the average error term in the usual approximation to the number of y-friable integers congruent to a modulo q, where a ≠ 0 is a fixed integer. We show that in the range exp{(log log x)5/3+ɛ} ⩽ yx and on average over qx/M with M → ∞ of moderate size, this average error term is asymptotic to −|a| Ψ(x/|a|, y)/2x. Previous results of this sort were obtained by the second author for reasonably dense sequences, however the sequence of y-friable integers studied in the current paper is thin, and required the use of different techniques, which are specific to friable integers.

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[1]Bourgain, J.. Decoupling, exponential sums and the Riemann zeta function. J. Amer. Math. Soc. à paraitre. 30 (2017), no. 1, 205224.
[2]de la Bretèche, R. et Tenenbaum, G.. Propriétés statistiques des entiers friables. Ramanujan Journal 9 (2005), n° 1-2, 139202.
[3]Drappeau, S.. Théorème de Fouvry–Iwaniec pour les entiers friables. Compositio Math. 151 (2015), pp. 828862.
[4]Fiorilli, D.. Residue classes containing an unexpected number of primes. Duke Math. J. 161 (2012), no. 15, 29232943.
[5]Fiorilli, D.. The influence of the first term of an arithmetic progression. Proc. London Math. Soc. (3) 106 (2013), no. 4, 819858.
[6]Fouvry, É. et Tenenbaum, G.. Répartition statistique des entiers sans grand facteur premier dans les progressions arithmétiques. Proc. London Math. Soc. (3) 72 (1996), no. 3, p. 481514.
[7]Harper, A. J.. On a paper of K. Soundararajan on smooth numbers in arithmetic progressions. J. Number Theory 132 (2012), no. 1, 182199.
[8]Harper, A. J.. Bombieri–Vinogradov and Barban–Davenport–Halberstam type theorems for smooth numbers, pré-publication (2012).
[9]Hildebrand, A.. Integers free of large prime factors and the Riemann hypothesis. Mathematika 31 (1984), no. 2, (1985), 258271.
[10]Hildebrand, A.. Integers free of large prime divisors in short intervals. Quart. J. Math. Oxford Ser. (2) 36 (1985), no. 141, 5769.
[11]Hildebrand, A.. On the number of positive integers ⩽ x and free of prime factors > y. J. Number Theory 22 (1986), 289–307.
[12]Hildebrand, A. et Tenenbaum, G.. On integers free of large prime factors. Trans. Amer. Math. Soc. 296 (1986), 265290.
[13]Lachand, A. et Tenenbaum, G.. Note sur les valeurs moyennes criblées de certaines fonctions arithmétiques. Quart. J. Math. (Oxford), 66 (2015), 245250.
[14]Saias, E.. Sur le nombre des entiers sans grand facteur premier. J. Number Theory 32 (1989), no. 1, 7899.
[15]Soundararajan, K.. The distribution of smooth numbers in arithmetic progressions, in: Anatomy of Integers, in: CRM Proc. Lect. Notes, vol. 46, Amer. Math. Soc. (Providence, RI, 2008), pp. 115128.
[16]Tenenbaum, G.. Introduction à la théorie analytique et probabiliste des nombres, troisième édition (coll. Échelles, Belin, 2008), 592 pp.
[17]Wolke, D.. Über die mittlere Verteilung der Werte zahlentheoretischer Funktionen auf Restklassen. I. Math. Ann. 202 (1973), p. 125.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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