On $\mathcal I(<q)$- and $\mathcal I(\leq q)$-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 $\mathcal I(<q)$, $\mathcal I(\leq q)$ of all subsets $A\subset \mathbb N$ with $\lambda(A)<q$, $\lambda(A)\le q$, respectively, satisfy $\mathcal I(<q)\subsetneq\mathcal I_c^{(q)}\subsetneq \mathcal I(\leq q)$, where $\mathcal I_c^{(q)}=\{A\subset\mathbb N: \sum_{a\in A}a^{-q}<\infty\}$. In this note we sharpen the results of Bal\'az, 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 $\mathcal I(\leq q)$-convergence, for which new methods and criteria are developed.