Translator Disclaimer
2021 Generalized lattice-point visibility in $\mathbb{N}^k$
Carolina Benedetti, Santiago Estupiñan, Pamela E. Harris
Involve 14(1): 103-118 (2021). DOI: 10.2140/involve.2021.14.103

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 = ( b 1 a 1 , b 2 a 2 , , b n a n ) ( { 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 1in 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.

Citation

Download Citation

Carolina Benedetti. Santiago Estupiñan. Pamela E. Harris. "Generalized lattice-point visibility in $\mathbb{N}^k$." Involve 14 (1) 103 - 118, 2021. https://doi.org/10.2140/involve.2021.14.103

Information

Received: 28 January 2020; Revised: 13 September 2020; Accepted: 28 September 2020; Published: 2021
First available in Project Euclid: 22 April 2021

Digital Object Identifier: 10.2140/involve.2021.14.103

Subjects:
Primary: 11B05
Secondary: 60B10

Keywords: generalized lattice-point visibility , lattice-point visibility , Riemann zeta function

Rights: Copyright © 2021 Mathematical Sciences Publishers

JOURNAL ARTICLE
16 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.14 • No. 1 • 2021
MSP
Back to Top