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Revision difference : $\tau$ function |
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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 {\sl $\tau$ function\/}, also called the {\sl divisor function\/}, takes positive integers 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 $\tau$ 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 $x$ is a nonnegative integer, then $\tau (p^x) = x+1$. |
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, then $\tau (4)= \tau (2^2)=2+1=3$. Since 2 and 5 are distinct primes, then $\tau (10)= \tau (2 \cdot 5)= \tau (2) \tau (5)=(1+1)(1+1)=4$. |
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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| The $\tau$ function is extremely useful for studying cyclic rings. |
The $\tau$ function is extremely useful for studying cyclic rings. |
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