On the Distribution of Rational Squares

Let $a$ be a positive integer, and let $\sigma (a)$ denote the least natural number $s$ such that an integer square lies between $s^{2}a$ and $s^2(a+1)$; let ${\tau }_s(a)$ denote the number of such integer squares. The function $\sigma (a)$ and the sequence $({\tau }_s(a){)}_{s\in {\mathbb{Z}}^{+}}$ are studied, and are observed to exhibit surprisingly chaotic behavior. Upper- and lower-bounds for $\sigma (a)$ are derived, as are criteria for when they are sharp.