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- W3134925230 abstract "It is a well-known result that the proportion of lattice points visible from the origin is given by $frac{1}{zeta(2)}$, where $zeta(s)=sum_{n=1}^inftyfrac{1}{n^s}$ denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika, generalized the notion of lattice point visibility by saying that for a fixed $binmathbb{N}$, a lattice point $(r,s)inmathbb{N}^2$ is $b$-visible from the origin if no other lattice point lies on the graph of a function $f(x)=mx^b$, for some $minmathbb{Q}$, between the origin and $(r,s)$. In their analysis they establish that for a fixed $binmathbb{N}$, the proportion of $b$-visible lattice points is $frac{1}{zeta(b+1)}$, which generalizes the result in the classical lattice point visibility setting. In this short note we give an $n$-dimensional notion of $bf{b}$-visibility that recovers the one presented by Goins et. al. in $2$-dimensions, and the classical notion in $n$-dimensions. We prove that for a fixed ${bf{b}}=(b_1,b_2,ldots,b_n)inmathbb{N}^n$ the proportion of ${bf{b}}$-visible lattice points is given by $frac{1}{zeta(sum_{i=1}^nb_i)}$. Moreover, we propose a $bf{b}$-visibility notion for vectors $bf{b}in mathbb{Q}_{>0}^n$, and we show that by imposing weak conditions on those vectors one obtains that the density of ${bf{b}}=(frac{b_1}{a_1},frac{b_2}{a_2},ldots,frac{b_n}{a_n})inmathbb{Q}_{>0}^n$-visible points is $frac{1}{zeta(sum_{i=1}^nb_i)}$. Finally, we give a notion of visibility for vectors $bf{b}in (mathbb{Q}^{*})^n$, compatible with the previous notion, that recovers the results of Harris and Omar for $bin mathbb{Q}^{*}$ in $2$-dimensions; and show that the proportion of $bf{b}$-visible points in this case only depends on the negative entries of $bf{b}$." @default.
- W3134925230 created "2021-03-15" @default.
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- W3134925230 date "2021-03-04" @default.
- W3134925230 modified "2023-10-14" @default.
- W3134925230 title "Generalized lattice-point visibility in ℕk" @default.
- W3134925230 cites W2325723526 @default.
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- W3134925230 doi "https://doi.org/10.2140/involve.2021.14.103" @default.
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