divisor function is multiplicative, the


The divisor functionDlmfDlmfMathworldPlanetmath (http://planetmath.org/TauFunction) is multiplicative.

Proof. Let t=mn with m,n coprimeMathworldPlanetmath. Applying the fundamental theorem of arithmeticMathworldPlanetmath, we can write


where each pj and qi are prime. Moreover, since m and n are coprime, we conclude that


Now, each divisorMathworldPlanetmathPlanetmath of t is of the form


with 0kjaj and 0hibi, and for each such divisor we get a divisor of m and a divisor of n, given respectively by


Now, each respective divisor of m, n is of the form above, and for each such pair their product is also a divisor of t. Therefore we get a bijection between the set of positive divisors of t and the set of pairs of divisors of m, n 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