product of divisors function

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}

If $n$ is not a square of an integer, $t$ is even (see http://planetmath.org/node/11781parity of $\tau$ function), whence

 $\displaystyle\begin{cases}a_{1}a_{t}\;=\;n\\ a_{2}a_{t-1}\;=\;n\\ \quad\cdots\\ a_{\frac{t}{2}}a_{\frac{t+2}{2}}\;=\;n.\end{cases}$

Thus

 $\prod_{d\mid n}d\;=\;a_{1}a_{2}\cdots a_{t}\;=\;n^{\frac{t}{2}}.$

If $n$ is a square of an integer, $t$ is odd, and we have

 $\displaystyle\begin{cases}a_{1}a_{t}\;=\;n\\ a_{2}a_{t-1}\;=\;n\\ \quad\cdots\\ a_{\frac{t-1}{2}}a_{\frac{t+3}{2}}\;=\;n\\ \;\;a_{\frac{t+1}{2}}\;=\;n^{\frac{1}{2}}.\end{cases}$

In this case we obtain a result:

 $\prod_{d\mid n}d\;=\;a_{1}a_{2}\cdots a_{t}\;=\;n^{\frac{t-1}{2}+\frac{1}{2}}% \;=\;n^{\frac{t}{2}}$

Title product of divisors function ProductOfDivisorsFunction 2013-03-22 18:55:45 2013-03-22 18:55:45 pahio (2872) pahio (2872) 7 pahio (2872) Definition msc 11A25 divisor product