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product of divisors function
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(Definition)
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The product of all positive divisors of a nonzero integer $n$ is equal $\sqrt{n^{\tau(n)}}$ , where tau function $\tau(n)$ expresses the number of the positive divisors of $n$ .
Proof. Let $t = \tau(n)$ and the positive divisors of $n$ be $a_1 < a_2 < \ldots < a_t.$
If $n$ is not a square of an integer, $t$ is even (see parity of $\tau$ function), whence
Thus $$\prod_{d \mid n}d \;=\; a_1a_2\cdots a_t \;=\; n^{\frac{t}{2}}.$$ If $n$ is a square of an integer, $t$ is odd, and we have
In this case we obtain a similar result: $$\prod_{d \mid n}d \;=\; a_1a_2\cdots a_t \;=\; n^{\frac{t-1}{2}+\frac{1}{2}} \;=\; n^{\frac{t}{2}}$$
Note. The absolute value of the product of all divisors is $n^{\tau(n)}.$
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"product of divisors function" is owned by pahio.
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| Other names: |
divisor product |
This object's parent.
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Cross-references: absolute value, odd, even, square, proof, number, tau function, integer, divisors, positive, product
This is version 4 of product of divisors function, born on 2009-05-14, modified 2009-05-20.
Object id is 11782, canonical name is ProductOfDivisorsFunction.
Accessed 793 times total.
Classification:
| AMS MSC: | 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas) |
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Pending Errata and Addenda
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