Mean value theorems for a class of density-like arithmetic functions

Abstract

This paper provides a mean value theorem for arithmetic functions [Formula: see text] defined by [Formula: see text] where [Formula: see text] is an arithmetic function taking values in [Formula: see text] and satisfying some generic conditions. As an application of our main result, we prove that the density [Formula: see text] (respectively, [Formula: see text]) of normal (respectively, primitive) elements in the finite field extension [Formula: see text] of [Formula: see text] are arithmetic functions of (nonzero) mean values.