A lattice point in ℤⁿ is visible from the origin if the line segment connecting them contains no other lattice point — equivalently, if the coordinates are coprime (their gcd is 1). The density of visible points in all of ℤⁿ is 1/ζ(n), where ζ is the Riemann zeta function. But what about visible points restricted to a hyperplane?
The paper on visible lattice points on hyperplanes (arXiv: 2603.22544) computes the asymptotic density: on the hyperplane a · x = b in ℝⁿ, the density of visible points is J_{n-1}(b) / b^{n-1}, where J_{n-1} is the Jordan totient function — the higher-dimensional generalization of Euler's totient.
The Jordan totient J_k(b) counts k-tuples of integers in {1,...,b} that, together with b, are coprime. It generalizes φ(b) = J_1(b). The formula says: the fraction of lattice points on the hyperplane that are visible depends only on b (the hyperplane's offset from the origin) and n (the ambient dimension), not on the normal direction a.
The results extend to intersections of hyperplanes and to k-th power-free points (points whose coordinates have no common k-th power factor). The closure of achievable densities for fixed dimension n is also determined.
The through-claim: visibility on a hyperplane depends on the hyperplane's arithmetic, not its geometry. The normal direction a is irrelevant — only the offset b matters. This is because visibility is a number-theoretic condition (coprimality), and the constraint a · x = b reduces the number theory to a single parameter. The geometry of the hyperplane is absorbed into the arithmetic of its defining equation.
2603.22544. Number theory / lattice points / visibility / Jordan totient function / hyperplanes.