# Liouville function

The *Liouville function ^{}* is defined by $\lambda (1)=1$ and $\lambda (n)={(-1)}^{{k}_{1}+{k}_{2}+\mathrm{\cdots}+{k}_{r}}$, if the prime factorization

^{}of $n>1$ is $n={p}_{1}^{{k}_{1}}{p}_{2}^{{k}_{2}}\mathrm{\cdots}{p}_{r}^{{k}_{r}}$ (where each ${p}_{i}$ is positive). This function

^{}is completely multiplicative and the

$$\sum _{d|n}\lambda (d)=\{\begin{array}{cc}1\hfill & \text{if}n={m}^{2}\text{for some integer}m\hfill \\ 0\hfill & \text{otherwise,}\hfill \end{array}$$ |

where the sum runs over all positive divisors^{} of $n$.

Title | Liouville function |
---|---|

Canonical name | LiouvilleFunction |

Date of creation | 2013-03-22 11:47:09 |

Last modified on | 2013-03-22 11:47:09 |

Owner | KimJ (5) |

Last modified by | KimJ (5) |

Numerical id | 12 |

Author | KimJ (5) |

Entry type | Definition |

Classification | msc 20G10 |

Classification | msc 11A25 |

Classification | msc 81-00 |