On I<q- and I≤q-convergence of arithmetic functions

Let N be the set of positive integers, and denote by λ ( A ) = inf { t > 0 : ∑ a ∈ A a - t < ∞ } the convergence exponent of A ⊂ N . For 0 < q ≤ 1 , 0 ≤ q ≤ 1 , respectively, the admissible ideals I < q , I ≤ q of all subsets A ⊂ N with λ ( A ) < q , λ ( A ) ≤ q , respectively, satisfy I < q ⊊ I c ( q ) ⊊ I ≤ q , where I c ( q ) = { A ⊂ N : ∑ a ∈ A a - q < ∞ } . In this note we sharpen the results of Baláž et al. from (J Number Theory 183:74–83, 2018) and other papers, concerning characterizations of I c ( q ) -convergence of various arithmetic functions in terms of q . This is achieved by utilizing I < q - and I ≤ q -convergence, for which new methods and criteria are developed.