## 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.

### MSC:

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