On the binomial convolution of arithmetical functions

TL;DR

This paper introduces and analyzes the binomial convolution of arithmetical functions, establishing its properties, algebraic structure, and connections to classical convolutions and inversion formulas.

Contribution

It defines the binomial convolution for arithmetical functions, explores its algebraic properties, and proves its isomorphism with the Dirichlet convolution algebra.

Findings

The binomial convolution forms a commutative algebra isomorphic to the Dirichlet convolution algebra.

Characterizations of multiplicative and Selberg multiplicative functions under this convolution.

Derived M"obius-type inversion formulas and a generalized multinomial theorem.

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.