Abstract
McCarthy [4] generalized a necessary and sufficient condition for an arithmetic function to be totally multiplicative to the incidence algebra on a partially ordered set. Several equivalent conditions for an arithmetic function to be totally multiplicative are known [1], [2]. In this paper we generalize several of these (and some apparently new ones) to the regular convolution rings of Narkiewicz [5]. We also investigate the prime factorization of arithmetic functions in a certain subring of some of these regular convolution rings.
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