Multiplicativity of the divisor function

Builds on Divisor function Multiplicative function

Statement

Let $k$ be a natural number. The divisor function $\sigma_k$ is a multiplicative function. That is, $\sigma_k(mn) = \sigma_k(m) \sigma_k(n)$ for all coprime positive integers $m$ and $n$ .

Remarks

The proof: for coprime $m$ and $n$ , each divisor of $mn$ is a unique product $d e$ with $d \mid m$ and $e \mid n$ . The function $\sigma_k$ is not completely multiplicative: $\sigma_0(4) = 3$ , but $\sigma_0(2)^2 = 4$ . At a prime power, $\sigma_k(p^e) = 1 + p^k + p^{2k} + \dots + p^{ek}$ . Multiplicativity then gives $\sigma_k(n)$ for all $n$ .

Reading path

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Arithmetic function definition
Divisibility definition
Divisors of a natural number definition
Greatest common divisor definition
Divisor function definition
Coprime integers definition
Multiplicative function definition
Multiplicativity of the divisor function statement

Tag	`04WC36XKV6`
URI	`m2://04WC36XKV6`
Kind	statement, theorem
Writing	W0 stub
MSC	11A25
Status	This copy of the site has no build data. Run `m2 status 04WC36XKV6`.
Uses	Divisor function, Multiplicative function

Multiplicativity of the divisor function

Statement

Remarks

Reading path

Further reading

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