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Zbl 1076.11051
Naimi, Mongi
Average distribution of certain arithmetic functions on the set of integers without large prime factors. (Répartition en moyenne de certaines fonctions arithmétiques sur l'ensemble des entiers sans grand facteur premier.) (French)
[J] J. Théor. Nombres Bordx. 15, No. 3, 745-766 (2003). ISSN 1246-7405

Let $\lambda> 1$, $0<\eta<{1\over 2}$ and $g(n)$ be a strictly positive multiplicative function satisfying $g(p)={1\over \lambda}$ for all primes $p$ and $g(n)\gg n^{-\eta}$ for all positive integers $n$, and let $f(n)= ng(n)$. Denote the largest prime divisor of $n\ge 2$ by $P(n)$ and put $P(1)= 1$. In this paper the author obtains an asymptotic formula for the sum $$S_f(x, y)= \sum\Sb f(n)\le x\\ P(n)\le y\endSb 1\tag1$$ that holds uniformly under the conditions $H_{\varepsilon, \lambda, c}: x\ge x_0$, $\exp((\log\log cx)^{{5\over 3}+ \varepsilon})\le{y\over\lambda}\le cx$, where $c= (\text{inf}\{f(n): n\ge 1\})^{-1}\ge 1$, and where $\log f(n)\ll\log n$ is assumed. The case $\lambda= 1$ was considered previously by the author [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 42, 147--164 (1999; Zbl 0963.11052)] and by {\it A. Smati} and {\it J. Wu} [Q. J. Math., Oxf. II. Ser. 50, No. 197, 111--130 (1999; Zbl 0923.11125)]. Using a known asymptotic formula for the sum on the right of (2) below, necessary and sufficient conditions for $$S_f(x,y)\sim \sum\Sb n\le x\\ P(n)\le y\endSb{1\over g(n)}\qquad (x\to\infty)\tag2$$ are deduced, and these take the form that $(x,y)\in H_{\varepsilon, 1,1}$ and $\lim_{x\to\infty} {\log x\log\log x\over (\log y)^2}= 0$. In [Acta Arith. 49, No. 4, 313--322 (1988; Zbl 0588.10046)], {\it R. Balasubramanian} and {\it K. Ramachandra} investigated these problems in the case when the condition $P(n)\le y$ is omitted. The asymptotic formulae for both the sums in (2) involve a generalization of the Dickman function. The proofs are intricate but elementary and depend on a Buchstab type identity and a key functional inequality for $S_f(x,y)$.
[Eira J. Scourfield (Egham)]

MSC 2000:: *11N25 Distribution of integers with specified multiplicative constraints
11N37 Asymptotic results on arithmetic functions

Keywords: multiplicative functions; integers without large prime factors; asymptotic formulae; equivalence of two sums

Citations: Zbl 0963.11052; Zbl 0923.11125; Zbl 0588.10046

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