Solutions to arithmetic convolution equations
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- by Helge Glöckner, Lutz G. Lucht and Štefan Porubský
- Proc. Amer. Math. Soc. 135 (2007), 1619-1629
- DOI: https://doi.org/10.1090/S0002-9939-07-08738-2
- Published electronically: January 4, 2007
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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 \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.References
- H. Biller, Analyticity and naturality of the multi-variable functional calculus, TU Darmstadt Preprint 2332, 2004; http://wwwbib.mathematik.tu-darmstadt.de/Math-Net /Preprints/Listen/files/2332.ps.gz
- Tim Carroll and A. A. Gioia, Roots of multiplicative functions, Compositio Math. 65 (1988), no. 3, 349–358. MR 932075
- Martin J. Cohen, Richard Stanley, and M. S. Klamkin, Problems and Solutions: Solutions of Advanced Problems: 5293, Amer. Math. Monthly 73 (1966), no. 5, 553–555. MR 1533818, DOI 10.2307/2315501
- Susanne Dierolf and Jochen Wengenroth, Inductive limits of topological algebras, Linear Topol. Spaces Complex Anal. 3 (1997), 45–49. Dedicated to Professor Vyacheslav Pavlovich Zahariuta. MR 1632483
- Paul-Olivier Dehaye, On the structure of the group of multiplicative arithmetical functions, Bull. Belg. Math. Soc. Simon Stevin 9 (2002), no. 1, 15–21. MR 1905645
- Klaus Floret, Lokalkonvexe Sequenzen mit kompakten Abbildungen, J. Reine Angew. Math. 247 (1971), 155–195 (German). MR 287271, DOI 10.1515/crll.1971.247.155
- Klaus Floret and Joseph Wloka, Einführung in die Theorie der lokalkonvexen Räume, Lecture Notes in Mathematics, No. 56, Springer-Verlag, Berlin-New York, 1968 (German). MR 0226355
- Helge Glöckner, Algebras whose groups of units are Lie groups, Studia Math. 153 (2002), no. 2, 147–177. MR 1948922, DOI 10.4064/sm153-2-4
- P. Haukkanen, Arithmetical equations involving semi-multiplicative functions and the Dirichlet convolution, Rend. Mat. Appl. (7) 8 (1988), no. 4, 511–517 (1989) (English, with Italian summary). MR 1032719
- Pentti Haukkanen, On the real powers of completely multiplicative arithmetical functions, Nieuw Arch. Wisk. (4) 15 (1997), no. 1-2, 73–77. MR 1470439
- Edwin Hewitt and J. H. Williamson, Note on absolutely convergent Dirichlet series, Proc. Amer. Math. Soc. 8 (1957), 863–868. MR 90680, DOI 10.1090/S0002-9939-1957-0090680-X
- John Knopfmacher, Abstract analytic number theory, North-Holland Mathematical Library, Vol. 12, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0419383
- Vichian Laohakosol, Divisors of some arithmetic functions, Proceedings of the Second Asian Mathematical Conference 1995 (Nakhon Ratchasima), World Sci. Publ., River Edge, NJ, 1998, pp. 139–151. MR 1660560
- Štefan Porubský, Structure of the group of quasi multiplicative arithmetical functions, Acta Acad. Paedagog. Agriensis Sect. Math. (N.S.) 30 (2003), 133–145. Dedicated to the memory of Professor Dr. Péter Kiss. MR 2054723
- David Rearick, Divisibility of arithmetic functions, Pacific J. Math. 112 (1984), no. 1, 237–248. MR 739148
- Wolfgang Schwarz, Ramanujan-Entwicklungen stark multiplikativer zahlentheoretischer Funktionen, Acta Arith. 22 (1972/73), 329–338 (German). MR 323740, DOI 10.4064/aa-22-3-329-338
- M. V. Subbarao, A class of arithmetical equations, Nieuw Arch. Wisk. (3) 15 (1967), 211–217. MR 223293
- Lucien Waelbroeck, Les algèbres à inverse continu, C. R. Acad. Sci. Paris 238 (1954), 640–641 (French). MR 73951
Bibliographic Information
- Helge Glöckner
- Affiliation: Fachbereich Mathematik, TU Darmstadt, Schlossgartenstraße 7, 64289 Darmstadt, Germany
- MR Author ID: 614241
- Email: gloeckner@mathematik.tu-darmstadt.de
- Lutz G. Lucht
- Affiliation: Institute of Mathematics, Clausthal University of Technology, Erzstraße 1, 38678 Clausthal-Zellerfeld, Germany
- Email: lucht@math.tu-clausthal.de
- Štefan Porubský
- Affiliation: Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 2, 18207 Prague 8, Czech Republic
- Email: Stefan.Porubsky@cs.cas.cz
- 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
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 1619-1629
- MSC (2000): Primary 11A25, 46H30
- DOI: https://doi.org/10.1090/S0002-9939-07-08738-2
- MathSciNet review: 2286069