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Remarques sur un théorème de G. Halász et A. Sárközy. (Remark on a theorem of G. Halász and A. Sárközy). (French) Zbl 0704.11029

Let E be an arbitrary set of primes, let \(\Omega_ E(n)\) denote the number of prime factors of n from the set E, counted with multiplicity, and denote by \(P_ x(\Omega_ E(n)=k)\) the proportion of positive integers \(n\leq x\) for which \(\Omega_ E(n)=k.\) G. Halász [Acta Math. Acad. Sci. Hung. 23, 425-432 (1972; Zbl 0255.10046)] and A. Sárközy [Period. Math. Hung. 8, 135-150 (1977; Zbl 0361.10043)] have given sharp uniform upper and lower bounds for \(P_ x(\Omega_ E(n)=k)\) for the range \(k\leq (2-\delta)E(x),\) where \(E(x)=\sum_{p\leq x,p\in E}1/p\) and \(\delta\) is any fixed positive number.
In the paper under review the author shows that for all k \[ P_ x(\Omega_ E(n)=k)\leq c_ 1p_ 1^{-k}\exp (c_ 2t-E_ 1(y))S_ k(p_ 1E_ 1(y)), \] where \(c_ 1\) and \(c_ 2\) are absolute constants, \(p_ 1=\min \{p :\) \(p\in E\}\), \(E_ 1=E\setminus \{p_ 1\}\), \(y=x/p_ 1\), \(t=\min (p_ 1,k/E_ 1(y))\), and \(S_ k(x)=\sum^{k}_{i=0}x^ i/i!\). He also gives a lower bound of similar form. These bounds improve on estimates of K. Norton [Enseign. Math., II. Ser. 28, 31-52 (1982; Zbl 0501.10046)], and in many cases are as sharp as those of Halász and Sárközy.
Reviewer: A.Hildebrand

MSC:

11N05 Distribution of primes
11N64 Other results on the distribution of values or the characterization of arithmetic functions
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References:

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