product of divisors function

The product of all positive divisorsMathworldPlanetmathPlanetmath of a nonzero integer n is equal nτ(n), where tau function τ(n) expresses the number of the positive divisors of n.

Proof.  Let  t=τ(n)  and the positive divisors of n be  a1<a2<<at.

If n is not a square of an integer, t is even (see of τ function), whence




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


In this case we obtain a result:


Note.  The absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the product of all divisors is nτ(n).

Title product of divisors function
Canonical name ProductOfDivisorsFunction
Date of creation 2013-03-22 18:55:45
Last modified on 2013-03-22 18:55:45
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Definition
Classification msc 11A25
Synonym divisor product