Exercise 4.3.17* (If $F = f \star 1$ is multiplicative, so is $f$)

Question

Suppose that $F(n) = \sum_{d \mid n} f(d)$ for all $n$. Show that if $F(n)$ is multiplicative then $f(n)$ is multiplicative.

Richard Ganaye · Accepted Answer

\begin{proof} 
Suppose that $F(n) = \sum_{d \mid n} f(d)$ for all $n$. By the Möbius inversion formula (Theorem 4.8),
$$f(n) = \sum_{d\mid n} \mu(d) F(n/d)$$
($f = \mu \star F$ is the Dirichlet product of $\mu$ and $F$).

Since $\mu$ and $F$ are multiplicative, by Problem 16, $f$ is also multiplicative.
\end{proof}

Exercise 4.3.17* (If $F = f \star 1$ is multiplicative, so is $f$)

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Exercise 4.3.17* (If $F = f \star 1$ is multiplicative, so is $f$)

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