Exercise 4.2.9 (Product of multiplicative functions)

Question

f $f(n)$ and $g(n)$ are multiplicative functions and $g(n) 
e 0$ for every $n$, show that the functions $F(n) = f(n) g(n)$ and $G(n) = f(n)/g(n)$ are also multiplicative.

Richard Ganaye · Accepted Answer

\begin{proof} 
For all positive integers $m,n$ such that $m\wedge n = 1$,
\begin{align*}
F(nm) &= f(nm) g(nm)\
&= f(n) f(m) g(n) g(m)\
&= F(n) F(m).
\end{align*}
Similarly
\begin{align*}
G(nm) &= \frac{f(nm)}{g(nm)}\
&= \frac{f(n) f(m)}{g(n)g(m)}\
&= G(n) G(m).
\end{align*}
If $f(n)$ and $g(n)$ are multiplicative functions and $g(n) 
e 0$ for every $n$, then $fg$ and $f/g$ are also multiplicative.
\end{proof}

Exercise 4.2.9 (Product of multiplicative functions)

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Exercise 4.2.9 (Product of multiplicative functions)

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