Cor. Mobius Inversion Formula for multiplicative functions.
Let $f,F$ be multiplicative functions such that $F(n)=\sum\limits_{d\mid n}f(d)$. Then $f(n)=\sum\limits_{d\mid n}\mu(\frac nd)F(d)$.
Proof.
Set:
- $\vec x = (f(e))_{e\mid n}$,
- $\vec a=(F(d))_{d\mid n}$
in the Mobius Inversion Theorem to get the result. $\quad\quad\square$
What is this notation supposed to represent? A matrix or a sequence? How would it look explicitly?: $\vec x = (f(e))_{e\mid n}$, $\vec a=(F(d))_{d\mid n}$