The main purpose of this note is to define a new arithmetic function relative to a fixed positive integer and to study some of its properties.

Let χ be a primitive Dirichlet character modulo k ≥ 3. In this paper, we define complex numbers associated with χ, which we denote by Cr(χ) (r = 0, 1,…), and we discuss their properties and their relationships with the generalized Bernoulli numbers.

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Let [Formula: see text] be a fixed integer, we define the multiplicative function [Formula: see text], where [Formula: see text] is the Piltz divisor function and [Formula: see text] is the unitary analogue function of [Formula: see text]. The main purpose of this paper to use elementary methods to study the mean value of the function [Formula: see text].

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