On a special class of \(A\)-functions (Q2280166)

An \(A\)-function associates with every positive integer \(n\) a subset \(A(n)\) of divisors of \(n\), and \[ f*_Ag(n) = \sum_{d\in A(n)}f(d)g(n/d) \] defines a convolution in the set \(\mathcal A\) of arithmetical functions. The authors study the connection of these objects with generalized divisibility relations introduced by \textit{G. Cohen} [Math. Comput. 54, 395--411 (1999; Zbl 0689.10014)], generalizing some results of a previous paper of the first two authors [Int. J. Number Theory 15, No. 9, 1771--1792 (2019; Zbl 1442.11010)] and characterize \(A\)-functions for which \(\mathcal A\) becomes a ring in which addition is defined by \(*_A\) and multiplication is taken pointwise. A characterization of the case when \(\mathcal A\) is a ring with usual addition and \(*_A\) is the multiplication has been given by the reviewer [Colloq. Math. 10, 81--94 (1963; Zbl 0114.26502)]. The authors show also that a part of their results can be extended to the more general convolutions \[ f*_Kg(n)=\sum_{d\mid n}K(n,d)f(d)g(n/d), \] (where \(K(x,y)\) is a complex-valued function) defined by \textit{T. M. K. Davison} [Can. Math, Bull. 9, 287--296 (1966; Zbl 0151.02804)].