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 $f(n)$.

Title	arithmetic function
Canonical name	ArithmeticFunction
Date of creation	2013-03-22 13:50:49
Last modified on	2013-03-22 13:50:49
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	10
Author	mathcam (2727)
Entry type	Definition
Classification	msc 11A25
Related topic	ConvolutionInversesForArithmeticFunctions
Related topic	PropertyOfCompletelyMultiplicativeFunctions
Related topic	DivisorSumOfAnArithmeticFunction
Defines	Dirichlet convolution