Number Theory

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Mobius Inversion, True/False

NUMTH-1RSWO1

Let $f$ and $g$ be arithmetic functions. Recall that the Mobius inversion formula states:

$$ g(n)=\sum_{d|n}f(d) \ \Longleftrightarrow \ f(n)=\sum_{d|n}\mu(d)g\left(\frac nd \right) $$

...where $\mu(n)$ is the Mobius function defined by:

$$ \mu(n):=\begin{cases} (-1)^r &\mbox{ if }n=p_1p_2 \cdots p_r \mbox{ for distinct primes }p_1, \ldots, p_r \\\ 0 &\mbox{ otherwise.}\end{cases} $$

Which of the following statements are correct?

Select ALL that apply.

A

If $f$ is multiplicative, then $g$ is also multiplicative.

B

If $g$ is multiplicative, then $f$ is also multiplicative.

C

If $f$ is multiplicative, then $g$ may not be multiplicative.

D

If $g$ is multiplicative, then $f$ may not be multiplicative.

E

If $f$ is not multiplicative, then $g$ is also not multiplicative.