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# $\tau$ 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 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.

## Mathematics Subject Classification

11A25*no label found*

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