arithmetic function

An arithmetic function is a function $f:{\mathbb{Z}}^{+}\to ℂ$ from the positive integers to the complex numbers.

Any algebraic function over ${\mathbb{Z}}^{+}$, as well as transcendental functions such as $\sin (n\pi )$ and $e^{n\pi i}$ with $n\in {\mathbb{Z}}^{+}$ 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

$(f+g)(n)=f(n)+g(n),$

and the Dirichlet convolution of $f$ and $g$, denoted by $f*g$, is given by

$(f*g)(n)={\displaystyle \sum _{d|n}}f(d)g\left({\displaystyle \frac{n}{d}}\right).$

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 $f(1)\ne 0$.

Note that giving a sequence $\{a_n\}$ of complex numbers is equivalent to giving an arithmetic function by associating $a_n$ with $f(n)$.