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Keywords:
Banach algebras; arithmetic functions; weighted norms; inversion; general Dirichlet series; Euler products
Banach algebras; arithmetic functions; weighted norms; inversion; general Dirichlet series; Euler products
Summary:
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.
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