An 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 sum of $f$ and $g$ , denoted $f+g$ , is given by \begin{align*} (f+g)(n)=f(n)+g(n), \end{align*}and the 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 $f(n)$ .