14. Below are various forms of the Möbius inversion formula, which can be directly verified or derived using $\sum_{d \mid n} \mu(d)=\left[\frac{1}{n}\right]$:
(i) Let $x \geqslant 1, K$ be a given positive integer, and let $1 \leqslant n \leqslant x, n \mid K$. Prove that $F(n) = \sum_{\substack{d|K \\ n| d \leqslant x}} f(d)$ holds if and only if $f(n) = \sum_{\substack{d|K \\ n| d \leqslant x}} \mu\left(\frac{d}{n}\right) F(d)$.
(ii) Let $\alpha(x), \beta(x)$ be functions defined on the interval $(0, \infty)$. Prove that $\beta(x) = \sum_{d=1}^{\infty} \alpha\left(\frac{x}{d}\right)$ holds if and only if $\alpha(x) = \sum_{d=1}^{\infty} \mu(d) \beta\left(\frac{x}{d}\right)$, assuming that for a given $x > 0$, the double series $\sum_{d=1}^{\infty} \sum_{k=1}^{\infty}\left|\alpha\left(\frac{x}{d k}\right)\right|$ and $\sum_{d=1}^{\infty} \sum_{k=1}^{\infty}\left|\beta\left(\frac{x}{d k}\right)\right|$ both converge.
(vi) Let $\alpha(x, y), \beta(x, y)$ be functions of two real variables $x, y$ defined on the rectangular region $0 < x_{0} \leqslant x \leqslant x_{1}, 0 < y_{0} \leqslant y \leqslant y_{1}$. Prove that
$$\beta(x, y) = \sum_{1 \leqslant d \leqslant x_{1} / x} \sum_{1 \leqslant l \leqslant y_{1} / y} \alpha(d x, l y)$$
holds if and only if
$$\alpha(x, y) = \sum_{1 \leqslant d \leqslant x_{1} / x} \sum_{1 \leqslant l \leqslant y_{1} / y} \mu(d) \mu(l) \beta(d x, l y).$$
(vii) Analogous to the generalization of (ii) to (iii), (iv), (v), make the corresponding generalization for (vi).