Geodesic
Banach algebra techniques in the theory of arithmetic functions
Communications in Mathematics, Tome 16 (2008) no. 1, pp. 45-56

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For infinite discrete additive semigroups $X\subset [0,\infty )$ we study normed algebras of arithmetic functions $g\colon X\rightarrow \mathbb {C}$ endowed with the linear operations and the convolution. In particular, we investigate the problem of scaling the mean deviation of related multiplicative functions for $X=\log {\mathbb {N}}$. This involves an extension of Banach algebras of arithmetic functions by introducing weight functions and proving a weighted inversion theorem of Wiener type in the frame of Gelfand’s theory of commutative Banach algebras.
Classification : 11N37, 11N56, 40E10, 46B25, 46J99
Keywords: Banach algebras; arithmetic functions; weighted norms; inversion; general Dirichlet series; Euler products 
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     author = {Lucht, Lutz G.},
     title = {Banach algebra techniques in the theory of arithmetic functions},
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     pages = {45--56},
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Lucht, Lutz G. Banach algebra techniques in the theory of arithmetic functions. Communications in Mathematics, Tome 16 (2008) no. 1, pp. 45-56. http://geodesic.mathdoc.fr/item/COMIM_2008__16_1_a4/