Ic(q)-convergence of arithmetical functions

Let n>1 be an integer with its canonical representation, n=p1α1⋅p2α2⋯pkαk. Put H(n)=max⁡{α1,…,αk}, h(n)=min⁡{α1,…,αk}, ω(n)=k, Ω(n)=α1+⋯+αk, f(n)=∏d|nd and f⁎(n)=f(n)n. Many authors deal with the statistical convergence of these arithmetical functions. For instance the notion of normal order is defined by means of statistical convergence. The statistical convergence is equivalent with Id-convergence, where Id is the ideal of all subsets of positive integers having the asymptotic density zero. In this paper we will study I-convergence of well known arithmetical functions, where I=Ic(q)={A⊆N:∑a∈Aa−q<+∞} is an admissible ideal on N for q∈(0,1〉 such that Ic(q)⊊Id.