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Solutions to arithmetic convolution equations

Authors: Helge Glöckner, Lutz G. Lucht and Stefan Porubsky
Journal: Proc. Amer. Math. Soc. 135 (2007), 1619-1629
MSC (2000): Primary 11A25, 46H30
Published electronically: January 4, 2007
MathSciNet review: 2286069
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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

$\displaystyle a_d*g^{*d}+a_{d-1}*g^{*(d-1)}+\cdots+a_1*g+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 $ 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.

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Additional Information

Helge Glöckner
Affiliation: Fachbereich Mathematik, TU Darmstadt, Schlossgartenstraße 7, 64289 Darmstadt, Germany

Lutz G. Lucht
Affiliation: Institute of Mathematics, Clausthal University of Technology, Erzstraße 1, 38678 Clausthal-Zellerfeld, Germany

Stefan Porubsky
Affiliation: Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 2, 18207 Prague 8, Czech Republic

Keywords: Arithmetic functions, Dirichlet convolution, polynomial equations, analytic equations, topological algebras, holomorphic functional calculus
Received by editor(s): February 10, 2006
Published electronically: January 4, 2007
Additional Notes: Research on this paper was begun while the second author visited the Academy of Sciences of the Czech Republic in Prague, partly supported by a travel grant of the Deutsche Forschungsgemeinschaft. He wishes to thank the Institute of Computer Science for their hospitality. The third author was supported by the Grant Agency of the Czech Republic, Grant# 201/04/0381, and by the Institutional Research Plan AV0Z10300504. For some of the period during which this work was carried out he enjoyed the hospitality and support of the Clausthal University of Technology.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2007 American Mathematical Society

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