The Lattice Points of a Circle
The Lattice Points of a Circle
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圆的格点
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通讯作者：
E. Landau
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作者：
G. Hardy;E. Landau
1.设r（x）表示正整数x可以表示为两个平方（正、负或零）之和的方式的数目，并且设R（x）=<$0 ≤ n ≤ x r（x）=<$0 ≤ p2 + q2 ≤ x1。因此，R（x）是在以原点为圆心、半径为x的圆的边界上或边界上的“格点”（坐标为p、q的点是正、负或零的整数）的数量。（1.1）R（x）− πx = O（x ½）是平凡的，哈代和朗道已经证明，关系R（x）− πx = O（x α）对于任何常数α <1/4都不成立，并且R（x）− πx = O（x 1/4）已经知道了一段时间。
1. Let r (x) denote the number of ways in which the positive integer x can be expressed as the sum of two squares (positive, negative or zero), and let R( x ) = Σ0 ≤ n ≤ x r(x) = Σ0 ≤ p 2 + q 2 ≤ x 1. Thus R( x ) is the number of “lattice-points” (points whose co-ordinate: p, q are integers, positive, negative or zero) in or on the boundary of the circle with centre at the origin and radius √ x . It is trivial that (1.1) R ( x ) − πx = O ( x ½), it has been shown by Hardy and Landau that the relation R ( x ) − πx = O ( x α ) is not true for any constant α < ¼, and it has been known for some time that R ( x ) − πx = O ( x ¼).