Banach algebra techniques in the theory of arithmetic functions
For infinite discrete additive semigroups we study normed algebras of arithmetic functions endowed with the linear operations and the convolution. In particular, we investigate the problem of scaling the mean deviation of related multiplicative functions for . 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.