Generalized lattice-point visibility in $\mathbb{N}^k$

A lattice point (r,s)∈ℕ2 is said to be visible from the origin if no other integer lattice point lies on the line segment joining the origin and (r,s). It is a well-known result that the proportion of lattice points visible from the origin is given by 1∕ζ(2), where ζ(s)= ∑n=1∞1∕ns denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika generalized the notion of lattice-point visibility by saying that for a fixed b∈ℕ a lattice point (r,s)∈ℕ2 is b-visible from the origin if no other lattice point lies on the graph of a function f(x)=mxb, for some m∈ℚ, between the origin and (r,s). In their analysis they establish that for a fixed b∈ℕ the proportion of b-visible lattice points is 1∕ζ(b+1), which generalizes the result in the classical lattice-point visibility setting. In this paper we give an n-dimensional notion of b-visibility that recovers the one presented by Goins et. al. in two dimensions, and the classical notion in n dimensions. We prove that for a fixed b=(b1,b2,…,bn)∈ℕn the proportion of b-visible lattice points is given by 1∕ζ(∑i=1nbi). Moreover, we give a new notion of b-visibility for vectors b = ( b 1 ∕ a 1 , b 2 ∕ a 2 , … , b n ∕ a n ) ∈ ( ℚ ∖ { 0 } ) n , with nonzero rational entries. In this case, our main result establishes that the proportion of b-visible points is 1∕ζ(∑i∈J|bi|), where J is the set of the indices 1≤i≤n for which bi∕ai<0. This result recovers a main theorem of Harris and Omar for b∈ℚ∖{0} in two dimensions, while showing that the proportion of b-visible points (in such cases) only depends on the negative entries of b.