arithmetic functionArithmeticFunction2003-08-14 15:19:132008-10-24 23:29:11DefinitionDirichlet convolution% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amsthm}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\mc}{\mathcal}
\newcommand{\mb}{\mathbb}
\newcommand{\mf}{\mathfrak}
\newcommand{\ol}{\overline}
\newcommand{\ra}{\rightarrow}
\newcommand{\la}{\leftarrow}
\newcommand{\La}{\Leftarrow}
\newcommand{\Ra}{\Rightarrow}
\newcommand{\nor}{\vartriangleleft}
\newcommand{\Gal}{\text{Gal}}
\newcommand{\GL}{\text{GL}}
\newcommand{\Z}{\mb{Z}}
\newcommand{\R}{\mb{R}}
\newcommand{\Q}{\mb{Q}}
\newcommand{\<}{\langle}
\renewcommand{\>}{\rangle}An \emph{arithmetic function} is a function $f:\Z^+\ra\mathbb{C}$ from the positive integers to the complex numbers.
Any algebraic function over $\Z^+$, as well as transcendental functions such as $\sin(n\pi)$ and $e^{n\pi i}$ with $n\in \Z^+$ are arithmetic functions.
There are two noteworthy operations on the set of arithmetic functions:
If $f$ and $g$ are two arithmetic functions, the \emph{sum} of $f$ and $g$, denoted $f+g$, is given by
\begin{align*}
(f+g)(n)=f(n)+g(n),
\end{align*}
and the \emph{Dirichlet convolution} of $f$ and $g$, denoted by $f*g$, is given by
\begin{align*}
(f*g)(n)=\sum_{d|n}f(d)g\left(\frac{n}{d}\right).
\end{align*}
The set of arithmetic functions, equipped with these two binary operations, forms a commutative ring with unity. The 0 of the ring is the function $f$ such that $f(n)=0$ for any positive integer $n$. The 1 of the ring is the function $f$ with $f(1)=1$ and $f(n)=0$ for any $n>1$, and the units of the ring are those arithmetic function $f$ such that $f(1)\neq 0$.
Note that giving a sequence $\{a_n\}$ of complex numbers is equivalent to giving an arithmetic function by associating $a_n$ with $f(n)$.