Divisor Sum Function is Multiplicative/Proof 1

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The divisor sum function:

$\ds {\sigma_1}: \Z_{>0} \to \Z_{>0}: \map {\sigma_1} n = \sum_{d \mathop \divides n} d$

is multiplicative.


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:

$\ds \map {\sigma_1} n = \sum_{d \mathop \divides n} d = \sum_{d \mathop \divides n} \map {I_{\Z_{>0} } } d$

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

The result follows from Sum Over Divisors of Multiplicative Function.