An arithmetic function f is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n. Furthermore an arithmetic function is said to be linear (or completely multiplicative) if f(1) = 1 and f(mn) = f(m)f(n) for all positive integers m and n.The Dirichlet convolution of two arithmetic functions f and g is defined by for all n∈Z+. Recall that the set of all multiplicative functions, denoted by M, with this operation is an abelian group.

No abstract

In this note we introduce a natural generalization of the ordinary convolution of arithmetic functions: If f and g are arithmetic functions,defines the K-product of f and g. If the kernel K(n) ≡ E(n) = 1, the K-product is the ordinary convolution Σd|nf(d)g(n/d);, if K(n) ≡ ∊(n) = [1/n], then the K-product is the unitary product Σf(d)g(n/d), summed over d|n, (d, n/d) = 1 (1, 2). We give in Theorem 1 a characterization of all associative kernels, i.e., kernels for which the corresponding K-product is associative.

No abstract

No abstract