Vol. 14, No. 1, 2021

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

Carolina Benedetti, Santiago Estupiñan and Pamela E. Harris

Vol. 14 (2021), No. 1, 103–118
Abstract

A lattice point (r,s) 2 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 1ζ(2), where ζ(s) = n=11ns denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika generalized the notion of lattice-point visibility by saying that for a fixed b a lattice point (r,s) 2 is b-visible from the origin if no other lattice point lies on the graph of a function f(x) = mxb , for some m , between the origin and (r,s). In their analysis they establish that for a fixed b the proportion of b-visible lattice points is 1ζ(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) n the proportion of b-visible lattice points is given by 1ζ( i=1nbi).

Moreover, we give a new notion of b-visibility for vectors

b = (b1a1,b2a2,,bnan) ( {0})n,

with nonzero rational entries. In this case, our main result establishes that the proportion of b-visible points is 1ζ( iJ|bi|), where J is the set of the indices 1 i n for which biai < 0. This result recovers a main theorem of Harris and Omar for b {0} in two dimensions, while showing that the proportion of b-visible points (in such cases) only depends on the negative entries of b.

Keywords
lattice-point visibility, generalized lattice-point visibility, Riemann zeta function
Mathematical Subject Classification 2010
Primary: 11B05
Secondary: 60B10
Milestones
Received: 28 January 2020
Revised: 13 September 2020
Accepted: 28 September 2020
Published: 4 March 2021

Communicated by Stephan Garcia
Authors
Carolina Benedetti
Departamento de Matemáticas
Universidad de los Andes
Bogotá
Colombia
Santiago Estupiñan
Departamento de Matemáticas
Universidad de los Andes
Bogotá
Colombia
Pamela E. Harris
Department of Mathematics and Statistics
Williams College
Williamstown, MA
United States