  In this article, we continue with our investigation of the Diophantine
equation $\frac{a}n=\frac1x+\frac1y$ and in particular its number of solutions
$R(n;a)$ for fixed $a$. We prove a couple of mean value theorems for the second
moment $(R(n;a))^2$ and from which we deduce $\log R(n;a)$ satisfies a certain
Gaussian distribution with mean $\log 3\log\log n$ and variance $(log
3)^2\log\log n$, which is an analog of the classical theorem of Erd\H os and
Kac. And finally these results in all suggest that the behavior of $R(n;a)$
resembles the divisor function $d(n^2)$ in various aspects.
