On a class of arithmetic convolutions involving arbitrary sets of integers. (English) Zbl 1026.11009

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.


11A25 Arithmetic functions; related numbers; inversion formulas
11N37 Asymptotic results on arithmetic functions
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