Florez 2019 distribution generalized greatest common divisor visibility

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corollary_5: Flórez, Karabulut and Quintero Vanegas's corollary that any b-pattern of crosses with a single circle is realizable, so isolated b-visible points exist and the graph G_b is disconnected; the paper also records Vardi's infinite component for G_1 and transfers it to G_b without further proof.

theorem_2: Flórez, Karabulut and Quintero Vanegas's mean-value theorem: for fixed b and an arithmetic function f with (1/N) times the sum over k <= N of |(f * mu)(k)|/k tending to 0, the mean of f(gcd_b(r,s)) over the lattice exists and equals zeta_f(b+1)/zeta(b+1) when zeta_f converges absolutely at b+1.

theorem_4: Flórez, Karabulut and Quintero Vanegas's density count: for fixed positive integers b and k, the proportion of lattice points (r,s) in N x N with gcd_b(r,s) = k is 1/(k^{b+1} zeta(b+1)); k = 1 gives the density 1/zeta(b+1) of b-visible points.

theorem_5: Flórez, Karabulut and Quintero Vanegas's average of the generalized gcd: for b >= 2 the sum of gcd_b(r,s) over 0 < r <= x and 0 < s <= x^b is x^{b+1} zeta(b)/zeta(b+1) + O(E(x)), with E(x) = x^2 log x for b = 2 and E(x) = x^b for b > 2.

theorem_6: Flórez, Karabulut and Quintero Vanegas's extension of Herzog and Stewart's pattern theorem: for fixed b > 1, a w by w^b pattern of prescribed b-visible and b-invisible points occurs in N x N exactly when its b-visible points contain no complete residue system modulo (p, p^b) for any prime p.

theorem_7: Flórez, Karabulut and Quintero Vanegas's mean value of a bounded function on the lattice: its mean equals zeta_{Lambda,b}(b+1)/zeta(b+1), where the k-th coefficient of zeta_{Lambda,b} is the average of Lambda over the points with gcd_b = k, and the series converges at b+1.

theorem_8: Flórez, Karabulut and Quintero Vanegas's neighbor count: the number of b-visible points of N x N at l^1 distance 1 from a point (r,s) has mean value 4/zeta(b+1) over the lattice, which the paper reads as G_b being 4/zeta(b+1)-connected on average.

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Jorge Flórez, Cihan Karabulut, Elkin Quintero Vanegas, The distribution of the generalized greatest common divisor and visibility of lattice points. Integers 20 (2020), #A6. arXiv:2002.10056. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2002.10056), every other right reserved. Labels and pages cited here are those of arXiv:2002.10056v1 (15 pages, dated 24 February 2020), the edition read, whose running head reads "INTEGERS: 19 (2019)".

The paper studies the generalized gcd of Goins, Harris, Kubik and Mbirika, $\gcd_b(r,s)=\max\{k: k\mid r,\ k^b\mid s\}$ (Definition 1, p. 2), whose value $1$ characterizes $b$-visibility of a point of $L=\mathbb N\times\mathbb N$. Theorem 2 (pp. 2--3) gives the mean value $\zeta_f(b+1)/\zeta(b+1)$ of $f(\gcd_b(r,s))$ over $L$ for $f$ with $N^{-1}\sum_{k\le N}\lvert(f*\mu)(k)\rvert/k\to0$ (for example $f$ bounded) and $\zeta_f$ absolutely convergent at $b+1$. Theorem 4 (p. 3) follows: the points with $\gcd_b=k$ have density $1/(k^{b+1}\zeta(b+1))$, and $k=1$ recovers the density $1/\zeta(b+1)$ of $b$-visible points. Theorem 5 (p. 3) gives, for $b\ge2$, the sum of $\gcd_b(r,s)$ over $0<r\le x$, $0<s\le x^b$ as $x^{b+1}\zeta(b)/\zeta(b+1)+O(E(x))$, with $E(x)=x^2\log x$ for $b=2$ and $x^b$ for $b>2$, without secondary terms. Theorem 7 (p. 7) writes the mean of a bounded function on $L$ through the Dirichlet series of its averages on the sets $\gcd_b=k$.

Section 3 (pp. 10--14) turns to the graph $G_b$ on the $b$-visible points, with edges at Euclidean distance $1$. Theorem 8 (p. 10) shows that a point of $L$ has on average $4/\zeta(b+1)$ $b$-visible neighbors. Theorem 6 (p. 4), proved as Theorem 11 (p. 12), characterizes for $b>1$ the realizable $b$-patterns: a $w\times w^b$ arrangement of prescribed $b$-visible and $b$-invisible points occurs in $L$ exactly when its $b$-visible points contain no complete rectangle modulo $(p,p^b)$ for any prime $p$, extending Herzog and Stewart's case $b=1$. Corollary 5 (p. 13) gives $b$-visible points surrounded by $b$-invisible ones, so $G_b$ is not connected, with the example $(6001645,49747967748324)$ for $b=2$ (p. 14). On p. 11 the paper reports Vardi's theorems that $G_1$ has a unique infinite component of positive asymptotic density, and states without further argument that, since $G_1\subset G_b$, $G_b$ for $b\ge2$ has a unique infinite component $C_b$ whose share of the square $\{1,\ldots,N\}^2$ is bounded below by a positive constant for large $N$. The statement of Corollary 6 (p. 14) prints "$N \geq 2^b$" [sic] where its proof derives $N<2^b$.

Source: https://arxiv.org/abs/2002.10056.

Read status: claims checked for Definitions 1, 9 and 10, Theorems 2, 4, 5, 6 (11), 7 and 8, Corollary 5 and the p. 11 discussion, read clause by clause on the print; the proofs were followed for structure and not checked independently. Vardi's results are cited, not proved, in the paper.

Bears on. #1212: the problem's graph of coprime pairs is the paper's $G_1$. Corollary 5 (p. 13) and the p. 11 discussion state that every $G_b$ with $b\ge1$ has isolated vertices, and report Vardi's unique infinite component of positive density for $G_1$, which the paper cites and does not prove; Theorem 8 (p. 10) averages the number of visible neighbors, and Theorem 6 extends Herzog and Stewart's pattern theorem, a reference of the problem, to $b>1$. The paper says nothing about paths avoiding points with a coordinate $1$ or points both of whose coordinates are prime.

Results. Pages are those of arXiv:2002.10056v1.

• Theorem 2 (pp. 2--3): mean value of $f(\gcd_b(r,s))$ over $L$ is $\zeta_f(b+1)/\zeta(b+1)$.
• Theorem 4 (p. 3): density $1/(k^{b+1}\zeta(b+1))$ of the points with $\gcd_b(r,s)=k$.
• Theorem 5 (p. 3): for $b\ge2$, the sum of $\gcd_b$ over the $x\times x^b$ box is $x^{b+1}\zeta(b)/\zeta(b+1)+O(E(x))$.
• Theorem 6 (p. 4; Theorem 11, p. 12): for $b>1$, a $b$-pattern is realizable iff its circles contain no complete rectangle modulo $(p,p^b)$ for any prime $p$.
• Theorem 7 (p. 7): for bounded $\Lambda$, $M(\Lambda)=\zeta_{\Lambda,b}(b+1)/\zeta(b+1)$.
• Theorem 8 (p. 10): a point of $L$ has on average $4/\zeta(b+1)$ $b$-visible neighbors.
• Corollary 5 (p. 13): isolated $b$-visible points exist, so $G_b$ is not connected; with the p. 11 report of Vardi's infinite component.

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