In this paper we investigate the ring $A_r(R)$ of arithmetical functions in r variables over an integral domain R with respect to the unitary convolution. We study a class of norms, and a class of derivations on $A_r(R)$. We also show that the resulting metric structure is complete.
"Unitary convolution for arithmetical functions in several variables." Hiroshima Math. J. 36 (1) 113 - 124, March 2006. https://doi.org/10.32917/hmj/1147883399