product of divisors function
The product of all positive divisors of a nonzero integer is equal , where tau function expresses the number of the positive divisors of .
Proof. Let and the positive divisors of be
If is not a square of an integer, is even (see http://planetmath.org/node/11781parity of function), whence
Thus
If is a square of an integer, is odd, and we have
In this case we obtain a result:
Note. The absolute value of the product of all divisors is