Upper bound of the local law for certain additive functions
type: Item
instance of: scholarly article
title: Upper bound of the local law for certain additive functions
zbMATH Open document ID: 0892.11027
DOI: 10.1007/BF01270610
author: Christian Tudesq
published in: Archiv der Mathematik
publication date: 1997-08-03T00:00:00Z
review text: Given a set \(E\) of primes, let \(\Omega_E(n)\) (resp. \(\omega_E(n))\) denote the number of prime factors of \(n\) belonging to \(E\), counted with (resp. without) multiplicity. \textit{G. Halász} [Acta Math. Sci. Hung. 23, 425-432 (1972; Zbl 0255.10046)] showed that, given a positive number \(\delta\), one has \[ \#\bigl\{n\leq x: \Omega_E(n)= k\bigr\} \ll_\delta x{e^{-E(x)} E(x)^k \over k!} \tag{1} \] with \(E(x)= \#\{p\leq x:p\in E\}\), uniformly in \(x\), \(E\), and (*) \(k\leq (1-\delta)E(x)\). The author shows that the same bound holds, without the restriction (*), for the function \(\omega_E(n)\). He also proves a similar, but slightly weaker bound for the joint distribution functions \[ \# \bigl\{n\leq x:\omega_{E_j} (n)=k_j\;(1\leq j\leq l) \bigr\}, \tag{2} \] where the sets \(E_j\) are pairwise disjoint sets of primes and the numbers \(k_j\) are arbitrary positive integers: There exist absolute positive constants \(\lambda\) and \(\mu\) such that the quantity (2) is bounded by \[ \leq\lambda x\exp \left\{-\sum^l_{j=1} E_j(x) \right\} \prod^l_{j=1} {(E_j(x) +\mu)^{k_j} \over k_j!},\tag{3} \] where \(E_j(x)= \#\{p\leq x:p \in E_j\}\). The author concludes by giving an example which shows that, in contrast to (1), the additive constant \(\mu\) in (3) is necessary in the case \(l\geq 2\).
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 1026275
zbMATH Keywords: upper bound
zbMATH Keywords: Hardy-Ramanujan theorem
zbMATH Keywords: additive functions
zbMATH Keywords: number of prime factors
zbMATH Keywords: joint distribution functions
reviewed by: Adolf Hildebrand
MaRDI profile type: Publication
cites work: Remarques sur un théorème de G. Halász et A. Sárközy
cites work: Remarks to my paper “on the distribution of additive and the mean values of multiplicative arithmetic functions”
cites work: On the number of prime factors of an integer
cites work: On the number of restricted prime factors of an integer. I
cites work: On the number of restricted prime factors of an integer. II
cites work: On the number of restricted prime factors of an integer. III
cites work: Remarks on a paper of G. Halasz
cites work: Q4863557
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description: scientific article; zbMATH DE number 1026275