# arithmetic function

An *arithmetic function ^{}* is a function

^{}$f:{\mathbb{Z}}^{+}\to \u2102$ from the positive integers to the complex numbers

^{}.

Any algebraic function^{} over ${\mathbb{Z}}^{+}$, as well as transcendental functions such as $\mathrm{sin}(n\pi )$ 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 |