$\tau$ 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.