# totient

A *totient* is a sequence $f:\{1,2,3,\mathrm{\dots}\}\to \u2102$ such
that

$$g\ast f=h$$ |

for some two completely multiplicative sequences $g$ and $h$, where $\ast $
denotes the convolution product^{} (or Dirichlet product; see multiplicative function).

The term ‘totient’ was introduced by Sylvester in the 1880’s, but is seldom used nowadays except in two cases. The Euler totient $\varphi $ satisfies

$${\iota}_{0}\ast \varphi ={\iota}_{1}$$ |

where ${\iota}_{k}$ denotes the function^{} $n\mapsto {n}^{k}$ (which is completely
multiplicative). The more general *Jordan totient* ${J}_{k}$ is defined by

$${\iota}_{0}\ast {J}_{k}={\iota}_{k}.$$ |

Title | totient |
---|---|

Canonical name | Totient |

Date of creation | 2013-03-22 13:38:35 |

Last modified on | 2013-03-22 13:38:35 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 11A25 |

Defines | totient |

Defines | Jordan totient |