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The {\sl $\tau$ function\/}, also called the {\sl 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, then $\tau (4)=3$. As another example, since 1, 2, 5, and 10 are all of the positive divisors of 10, then $\tau (10)=4$. |
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| The $\tau$ \emph{function}, also called the \emph{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 $\tau (10)=4$. |
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| The $\tau$ function behaves according to the following two rules: |
The $\tau$ function behaves according to the following two rules: |
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1. If $p$ is a prime and $k$ is a nonnegative integer, then $\tau(p^k)=k+1$.
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1. If $p$ is a prime and $x$ is a nonnegative integer, then $\tau (p^x) = x+1$.
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| 2. If $\gcd(a,b)=1$, then $\tau(ab)=\tau(a)\tau(b)$. |
2. If $\gcd(a,b)=1$, then $\tau(ab)=\tau(a)\tau(b)$. |
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| Because these two rules hold for the $\tau$ function, it is a multiplicative function. |
Because these two rules hold for the $\tau$ function, it is a multiplicative function. |
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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 $\tau(10)=\tau(2\cdot 5)=\tau(2)\tau(5)=(1+1)(1+1)=4$.
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Note that these rules work for the previous two examples. Since 2 is prime, then $\tau (4)= \tau (2^2)=2+1=3$. Since 2 and 5 are distinct primes, then $\tau (10)= \tau (2 \cdot 5)= \tau (2) \tau (5)=(1+1)(1+1)=4$.
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| If $n$ is a positive integer, the number of \PMlinkname{prime factors}{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 $\tau(9)=3$. |
If $n$ is a positive integer, the number of \PMlinkname{prime factors}{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 $\tau(9)=3$. |
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| The $\tau$ function is extremely useful for studying cyclic rings. |
The $\tau$ function is extremely useful for studying cyclic rings. |
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| The sequence $\{\tau(n)\}$ appears in the OEIS as sequence \PMlinkexternal{A000005}{http://www.research.att.com/~njas/sequences/A000005}. |
The sequence $\{\tau(n)\}$ appears in the OEIS as sequence \PMlinkexternal{A000005}{http://www.research.att.com/~njas/sequences/A000005}. |
