Lattice Point Visibility on Generalized Lines of Sight

For a fixed b ∈ N = {1, 2, 3, . . .} we say that a point (r, s) in the integer lattice Z × Z is b-visible from the origin if it lies on the graph of a power function f(x) = ax^b with a and no other integer lattice point lies on this curve (i.e., line of sight) between (0, 0) and (r, s). We prove that the proportion of b-visible integer lattice points is given by 1/ζ(b + 1), where ζ(s) denotes the Riemann zeta function. We also show that even though the proportion of b-visible lattice points approaches 1 as b approaches infinity, there exist arbitrarily large rectangular arrays of b-invisible lattice points for any fixed b. This work specialized to b = 1 recovers original results from the classical lattice point visibility setting where the lines of sight are given by linear functions with rational slope through the origin.