On the binomial convolution of arithmetical functions

Let n = ∏ p p νp(n) denote the canonical factorization of n ∈ N. The binomial convolution of arithmetical functions f and g is defined as (f ◦g)(n) = ∑ d|n (∏ p (νp(n) νp(d) )) f(d)g(n/d), where ( a b ) is the binomial coefficient. We provide properties of the binomial convolution. We study the Calgebra (A,+, ◦,C), characterizations of completely multiplicative functions, Selberg multiplicative functions, exponential Dirichlet series, exponential generating functions and a generalized binomial convolution leading to various Möbius-type inversion formulas. Throughout the paper we compare our results with those of the Dirichlet convolution. We also obtain a “multiplicative” version of the multinomial theorem. Mathematics Subject Classification: 11A25, 05Axx