Solutions to arithmetic convolution equations

Abstract

In the C-algebra A of arithmetic functions g: N → C, endowed with the usual pointwise linear operations and the Dirichlet convolution, let g *k denote the convolution power g*···*g with k factors g ∈ A. We investigate the solvability of polynomial equations of the form a d *g *d + a d-1 * g *(d-1) + ··· + a 1 * g + a 0 = 0 with fixed coefficients a d , a d-1 ,···, a 1 ,a 0 ∈ A. In some cases the solutions have specific properties and can be determined explicitly. We show that the property of the coefficients to belong to convergent Dirichlet series transfers to those solutions g ∈ A, whose values g(1) are simple zeros of the polynomial a d (1)z d + a d-1 (1)z d-1 + ···+ a 1 (1)z + a 0 (1). We extend this to systems of convolution equations, which need not be of polynomial-type.