Tóth, László On a class of arithmetic convolutions involving arbitrary sets of integers. (English) Zbl 1026.11009 Math. Pannonica 13, No. 2, 249-263 (2002). Let \(d\), \(n\) be positive integers and \(S\) be an arbitrary set of positive integers. We say that \(d\) is an \(S\)-divisor of \(n\) if \(d|n\) and \(\gcd(d,n/d) \in S\), and denote it by \(d|_sn\). Consider the \(S\)-convolution of arithmetical functions \(f\) and \(g\) defined by \[ (f*_sg)(n)= \sum_{d|_sn}f(d)g(n/d)=\sum_{d|n}\rho_s((d, n/d))f(d)g(n/d), \] where \(\rho_s\) stands for the characteristic function of \(S\). In the paper the set \(S\) is determined such that the \(S\)-convolution is associative and preserves the multiplicativity of functions. Asymptotic formulae are given with error terms for the functions \(\sigma_S(n)\) and \(\tau_S(n)\), representing the sum and the number of \(S\)-divisors of \(n\), respectively, for an arbitrary \(S\). The results generalize, unify and sharpen previous ones. Reviewer: Ferenc Mátyás (Eger) MSC: 11A25 Arithmetic functions; related numbers; inversion formulas 11N37 Asymptotic results on arithmetic functions Keywords:arithmetic convolution; characteristic function; multiplicative function; completely multiplicative function; divisor function; Möbius function; asymptotic formula; maximal order PDFBibTeX XMLCite \textit{L. Tóth}, Math. Pannonica 13, No. 2, 249--263 (2002; Zbl 1026.11009) Full Text: arXiv EuDML