For infinite discrete additive semigroups $X\subset [0,\infty )$ we study normed algebras of arithmetic functions $g:X\to ℂ$ 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.