On the binomial convolution of arithmetical functions

Abstract

Let $n=\prod_p p^{\nu_p(n)}$ denote the canonical factorization of $n\in \N$. The binomial convolution of arithmetical functions $f$ and $g$ is defined as $(f\circ g)(n)=\sum_{d\mid n} (\prod_p \binom{\nu_p(n)}{\nu_p(d)}) f(d)g(n/d),$ where $\binom{a}{b}$ is the binomial coefficient. We provide properties of the binomial convolution. We study the $\C$-algebra $({\cal A},+,\circ,\C)$, characterizations of completely multiplicative functions, Selberg multiplicative functions, exponential Dirichlet series, exponential generating functions and a generalized binomial convolution leading to various M\"obius-type inversion formulas. Throughout the paper we compare our results with those of the Dirichlet convolution *. Our main result is that $({\cal A},+,\circ,\C)$ is isomorphic to $({\cal A},+,*,\C)$. We also obtain a "multiplicative" version of the multinomial theorem.