divisor function is multiplicative, the
Theorem.
The divisor function (http://planetmath.org/TauFunction) is multiplicative.
Proof.
Let with coprime.
Applying the fundamental theorem of arithmetic
, we can write
where each and are prime. Moreover, since and are coprime, we conclude that
Now, each divisor of is of the form
with and , and for each such divisor we get a divisor of and a divisor of , given respectively by
Now, each respective divisor of , is of the form above, and for each such pair their product is also a divisor of . Therefore we get a bijection between the set of positive divisors of and the set of pairs of divisors of , respectively. Such bijection implies that the cardinalities of both sets are the same, and thus
Title | divisor function is multiplicative, the |
---|---|
Canonical name | DivisorFunctionIsMultiplicativeThe |
Date of creation | 2013-03-22 15:03:47 |
Last modified on | 2013-03-22 15:03:47 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 9 |
Author | yark (2760) |
Entry type | Theorem |
Classification | msc 11A25 |