On the binomial convolution of arithmetical functions

الأصل · EN

Let n=∏ₚ pνᵖ⁽ⁿ⁾ denote the canonical factorization of 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 ab is the binomial coefficient. We provide properties of the binomial convolution. We study the -algebra (A,+,∘,), 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 *. Our main result is that (A,+,∘,) is isomorphic to (A,+,*,). We also obtain a "multiplicative" version of the multinomial theorem.