On the binomial convolution of arithmetical functions

Abstract

Let $n={\prod }_p{p}^{{\nu }_p(n)}$ denote the canonical factorization of $n\in \mathbb{N}$. The binomial convolution of arithmetical functions $f$ and $g$ is defined as $(f\circ g)(n)={\sum }_{d\mid n}({\prod }_p(\genfrac{}{}{0ex}{}{{\nu }_p(n)}{{\nu }_p(d)}))f(d)g(n/d),$ where $\left(\genfrac{}{}{0ex}{}{a}{b}\right)$ 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 Möbius-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.

15 pages, revised

Cited by in corpus (2)

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