$\tau$ 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 $\gcd (a,b)=1$, 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