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Number Theory

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Completely Multiplicative Functions

NUMTH-6NH7VT

An arithmetic function $f$ is multiplicative if $f(mn)=f(m)f(n)$ for coprime $m$, $n$. We say that a multiplicative arithmetic function $g$ is completely multiplicative if $g(mn)=g(m)g(n)$ for all $m$, $n$.

For the following functions, find multiplicative but not completely multiplicative functions.

Check ALL that apply. The letter $p$ denote the prime numbers, and $p|n$ means $p$ divides $n$.

A

$f(n)=n^2$.

B

$f(n)=\sum_{p|n} 1$ .

C

$f(n)=n^2-n$.

D

$f(n)=e^n$.

E

$f(n)=\prod_{p|n} p$.