论文标题
广义最大的共同除数的分布和晶格点的可见性
The distribution of the generalized greatest common divisor and visibility of lattice points
论文作者
论文摘要
对于固定的$ b \ in \ mathbb {n} = \ {1,2,3，\ dots \} $，Goins等。 \ cite {harris}定义了$ b $ - 可视性的概念$ l = \ l = \ mathbb {n} \ times \ times \ mathbb {n} $的概念，该$ b $是$ b $ viss $ b $ viss ot onary the $ b $，如果它在$ b $上，如果$ b $ vis of the $ by-viss of of the $ for of the $ for a ax $ f ax） \ mathbb {q} $，$ l $中没有其他晶格点位于$（0,0）$和$（r，s）$之间。此外，要研究$ b $可见点的密度，$ l $ goins等人。定义了最大的普通分裂的概括，用$ \ gcd_b $表示，并证明了$ l $中$ b $可见的晶格点的比例由$ 1/ζ（b+1）$给出，其中$ζ（s）$是riemann zeta函数。在本文中，我们研究了使用$ \ gcd_b $定义的算术函数的平均值$λ：l \ to \ mathbb {c} $，并恢复了本文更一般结果的\ cite {harris}的主要结果。我们还调查了\ cite {harris}的结果的概括，该结果断言，在固定$ b $的晶格$ l $中，任意大型矩形安排为固定的$ b $，更具体地说，我们为$ b $ $ b $ $ b $ b $ boint的矩形安排提供了必要和足够的条件。我们的结果灵感来自Herzog和Stewart \ Cite {Herzog}的工作，后者在$ b = 1 $的情况下证明了这一点。
For a fixed $b\in \mathbb{N}=\{1,2,3,\dots\}$, Goins et al. \cite{Harris} defined the concept of $b$-visibility for a lattice point $(r,s)$ in $L=\mathbb{N}\times \mathbb{N}$ which states that $(r,s)$ is $b$-visible from the origin if it lies on the graph of $f(x)=ax^b$, for some positive $a\in \mathbb{Q}$, and no other lattice point in $L$ lies on this graph between $(0,0)$ and $(r,s)$. Furthermore, to study the density of $b$-visible points in $L$ Goins et al. defined a generalization of greatest common divisor, denoted by $\gcd_b$, and proved that the proportion of $b$-visible lattice points in $L$ is given by $1/ζ(b+1)$, where $ζ(s)$ is the Riemann zeta function. In this paper we study the mean values of arithmetic functions $Λ:L\to \mathbb{ C}$ defined using $\gcd_b$ and recover the main result of \cite{Harris} as a consequence of the more general results of this paper. We also investigate a generalization of a result in \cite{Harris} that asserts that there are arbitrarily large rectangular arrangements of $b$-visible points in the lattice $L$ for a fixed $b$, more specifically, we give necessary and sufficient conditions for an arbitrary rectangular arrangement containing $b$-visible and $b$-invisible points to be realizable in the lattice $L$. Our result is inspired by the work of Herzog and Stewart \cite{Herzog} who proved this in the case $b=1$.
