Divisor Sum Function is Multiplicative/Proof 1
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Theorem
The divisor sum function:
- $\ds {\sigma_1}: \Z_{>0} \to \Z_{>0}: \map {\sigma_1} n = \sum_{d \mathop \divides n} d$
is multiplicative.
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:
- $\ds \map {\sigma_1} n = \sum_{d \mathop \divides n} d = \sum_{d \mathop \divides n} \map {I_{\Z_{>0} } } d$
But from Identity Function is Completely Multiplicative, $I_{\Z_{>0} }$ is multiplicative.
The result follows from Sum Over Divisors of Multiplicative Function.
$\blacksquare$