BASIC PROPERTY OF GENERALIZED COMPLETELY MULTIPLICATIVE FUNCTIONS

Abstract

An arithmetic function f is said to be completely multiplicative if

f (1) = 1 and f (mn) = f (m)f (n) for all positive integers m and n.

In this paper, we define that an arithmetic function f is a generalized

completely multiplicative function if f (1) = 1 and there is a completely

multiplicative function fb such that f (mn) = f (m)fb (n)f (n)fb (m) for all

positive integers m and n. We consider some basic structure properties of

these functions. The functions v(n) = nn and ExpD are examples of gen-

eralized completely multiplicative functions, where D is the arithmetic

derivative.