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Solutions to arithmetic convolution equations

Authors:	Helge Glö ckner Lutz G Lucht Stefan Porubsky

Institution:	Fachbereich Mathematik, TU Darmstadt, Schlossgartenstraße 7, 64289 Darmstadt, Germany ; Institute of Mathematics, Clausthal University of Technology, Erzstraße 1, 38678 Clausthal-Zellerfeld, Germany ; Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou vezí 2, 18207 Prague 8, Czech Republic

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 $g\cdotsg$ with factors $g\in\mathscr{A}$ . We investigate the solvability of polynomial equations of the form $\displaystyle a_dg^{d}+a_{d-1}g^{(d-1)}+\cdots+a_1g+a_0=0$ 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 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.

Keywords:	Arithmetic functions Dirichlet convolution polynomial equations analytic equations topological algebras holomorphic functional calculus

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