On $\mathcal{I}(<q)$- and $\mathcal{I}(\le q)$-convergence of arithmetic functions

Abstract

Let $\mathbb{N}$ be the set of positive integers, and denote by $\lambda (A)=\inf \{t>0:{\sum }_{a\in A}{a}^{-t}<\infty \}$ the convergence exponent of $A\subset \mathbb{N}$. For $0<q\le 1$, $0\le q\le 1$, respectively, the admissible ideals $\mathcal{I}(<q)$, $\mathcal{I}(\le q)$ of all subsets $A\subset \mathbb{N}$ with $\lambda (A)<q$, $\lambda (A)\le q$, respectively, satisfy $\mathcal{I}(<q)⊊{\mathcal{I}}_c^{(q)}⊊\mathcal{I}(\le q)$, where ${\mathcal{I}}_c^{(q)}=\{A\subset \mathbb{N}:{\sum }_{a\in A}{a}^{-q}<\infty \}$. In this note we sharpen the results of Baláz, Gogola and Visnyai from [2], and of others papers, concerning characterizations of ${\mathcal{I}}_c^{(q)}$-convergence of various arithmetic functions in terms of $q$. This is achieved by utilizing $\mathcal{I}(<q)$- and $\mathcal{I}(\le q)$-convergence, for which new methods and criteria are developed.