THE Qα-CONVOLUTION OF ARITHMETIC FUNCTIONS AND SOME OF ITS PROPERTIES

Abstract

Let α be an arithmetic function such that α(n) =0(n ∈ N). The

Q

α

-convolution of two arithmetic functions is defined as

(f g)(n) =

ij=n

α( α i) ( α n ( ) j) f(i)g(j).

Basic properties of the Q

α

-convolution and characterizations of completely

multiplicative functions using Q

α

-convolution are derived. The solubility

of the equation

Tαg := ad g d + ad−1 g (d−1) + ··· + a1 g + a0 = 0

with fixed arithmetic functions ad(= 0), ad−1,...,a1, a0 is investigated.