  Counting functions are constructed for sums of integers raised to a fixed
positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k}
+ ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or
equal to a given $w>0$ is determined. The counting functions developed are
framed in terms of convolution exponentials, and are closely related to the
Riemann zeta function. At the conclusion, several estimates are derived, with
special emphasis on the case of sums of square roots, i.e. $j=1$, $k=2$.
