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). TheQ α-convolution of two arithmetic functions is defined as (f g)(n)= ij=n α(n) α(i)α(j) f(i)g(j).
Basic properties of the Qα-convolution and characterizations of completely multiplicative functions using Qα-convolution are derived. The solubility of the equation Tαg := ad gd + ad−1 g(d−1) +···+ a1 g + a0 =0 with fixed arithmetic functions ad(= 0),ad−1,...,a1,a0 is investigated.