Divisor Sum Function is Multiplicative/Proof 1

Proof
Let $I_{\Z_{>0}}: \Z_{>0} \to \Z_{>0}$ be the identity function:


 * $\forall n \in \Z_{>0}: \map {I_{\Z_{>0} } } n = n$

Thus we have:


 * $\displaystyle \map \sigma n = \sum_{d \mathop \divides n} d = \sum_{d \mathop \divides n} \map {I_{\Z_{>0} } } d$

But from Identity Function is Completely Multiplicative, $I_{\Z_{>0} }$ is multiplicative.

The result follows from Sum Over Divisors of Multiplicative Function.