# $\tau$ 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 tauFunction 2013-03-22 13:30:16 2013-03-22 13:30:16 Wkbj79 (1863) Wkbj79 (1863) 21 Wkbj79 (1863) Definition msc 11A25 divisor function Divisor DirichletHyperbolaMethod 2omeganLeTaunLe2Omegan Divisibility ValuesOfNForWhichVarphintaun LambertSeries ParityOfTauFunction