# normal order

Let $f(n)$ and $F(n)$ be functions from ${\mathbb{Z}}^{+}\to \mathbb{R}$. We say that $f(n)$ has normal order $F(n)$ if for each $\u03f5>0$ the set

$$ |

has the property that $\underset{\xaf}{d}(A(\u03f5))=1$. Equivalently, if $B(\u03f5)={\mathbb{Z}}^{+}\backslash A(\u03f5)$, then $\underset{\xaf}{d}(B(\u03f5))=0$. (Note that $\underset{\xaf}{d}(X)$ denotes the lower asymptotic density of $X$).

We say that $f$ has average order $F$ if

$$\sum _{j=1}^{n}f(j)\sim \sum _{j=1}^{n}F(j)$$ |

Title | normal order |
---|---|

Canonical name | NormalOrder |

Date of creation | 2013-03-22 12:36:23 |

Last modified on | 2013-03-22 12:36:23 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 11B05 |

Defines | average order |