Upper bound of the local law for certain additive functions
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 \(l\geq 2\).
scientific article; zbMATH DE number 1026275
scientific article