# arithmetic function

An arithmetic function is a function $f:\mathbb{Z}^{+}\rightarrow\mathbb{C}$ 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

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

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

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

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 $f(1)\neq 0$.

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

Title arithmetic function ArithmeticFunction 2013-03-22 13:50:49 2013-03-22 13:50:49 mathcam (2727) mathcam (2727) 10 mathcam (2727) Definition msc 11A25 ConvolutionInversesForArithmeticFunctions PropertyOfCompletelyMultiplicativeFunctions DivisorSumOfAnArithmeticFunction Dirichlet convolution