Solutions to arithmetic convolution equations

Abstract:

In the $\mathbb {C}$-algebra $\mathscr {A}$ of arithmetic functions $g\colon \mathbb N\to \mathbb {C}$, endowed with the usual pointwise linear operations and the Dirichlet convolution, let $g^{*k}$ denote the convolution power $g*\cdots *g$ with $k$ factors $g\in \mathscr {A}$. We investigate the solvability of polynomial equations of the form \begin{equation*} a_d*g^{*d}+a_{d-1}*g^{*(d-1)}+\cdots +a_1*g+a_0=0 \end{equation*} with fixed coefficients $a_d,a_{d-1},\ldots ,a_1,a_0\in \mathscr {A}$. 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 $g\in \mathscr {A}$, whose values $g(1)$ are simple zeros of the polynomial $a_d(1)z^d+a_{d-1}(1)z^{d-1}+\cdots +a_1(1)z+a_0(1)$. We extend this to systems of convolution equations, which need not be of polynomial-type.