# $\tau $ function

The $\tau $ *function ^{}*, also called the

*divisor function*, takes a positive integer as its input and gives the number of positive divisors

^{}^{}of its input as its output. For example, since $1$, $2$, and $4$ are all of the positive divisors of $4$, we have $\tau (4)=3$. As another example, since $1$, $2$, $5$, and $10$ are all of the positive divisors of $10$, we have $\tau (10)=4$.

The $\tau $ function behaves according to the following two rules:

1. If $p$ is a prime and $k$ is a nonnegative integer, then $\tau ({p}^{k})=k+1$.

2. If $\mathrm{gcd}(a,b)=1$, then $\tau (ab)=\tau (a)\tau (b)$.

Because these two rules hold for the $\tau $ function, it is a multiplicative function^{}.

Note that these rules work for the previous two examples. Since $2$ is prime, we have $\tau (4)=\tau ({2}^{2})=2+1=3$. Since $2$ and $5$ are distinct primes, we have $\tau (10)=\tau (2\cdot 5)=\tau (2)\tau (5)=(1+1)(1+1)=4$.

If $n$ is a positive integer, the number of prime factors^{} (http://planetmath.org/UFD) of ${x}^{n}-1$ over $\mathbb{Q}[x]$ is $\tau (n)$. For example, ${x}^{9}-1=({x}^{3}-1)({x}^{6}+{x}^{3}+1)=(x-1)({x}^{2}+x+1)({x}^{6}+{x}^{3}+1)$ and $\tau (9)=3$.

The $\tau $ function is extremely useful for studying cyclic rings.

The sequence $\{\tau (n)\}$ appears in the OEIS as sequence http://www.research.att.com/ njas/sequences/A000005A000005.

Title | $\tau $ function |
---|---|

Canonical name | tauFunction |

Date of creation | 2013-03-22 13:30:16 |

Last modified on | 2013-03-22 13:30:16 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 21 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 11A25 |

Synonym | divisor function |

Related topic | Divisor |

Related topic | DirichletHyperbolaMethod |

Related topic | 2omeganLeTaunLe2Omegan |

Related topic | Divisibility |

Related topic | ValuesOfNForWhichVarphintaun |

Related topic | LambertSeries |

Related topic | ParityOfTauFunction |