【作者】 陈鑫；

【导师】 任秀敏；

【作者基本信息】 山东大学 ， 基础数学， 2016， 硕士

【摘要】 整点问题是数论中的一类重要问题,Gauss和Dirichlet最先研究了这类问题并提出了两个著名的论断,即“关于圆内整点个数的Gauss问题”和"Dirichlet除数问题”.设C(R)表示圆x2+y2≤R内的整点个数,Gauss证明了之后,很多数学家([8,9,13,14,16,21])研究了这一问题并对上式中的余项做出了改进.1993年,Huxley[9]将上式中的余项改进至O(R23/73+ε).之后,HuX-ley[10]又将(1)式中的余项改进至O(R131/416+ ε)，这是目前最好的结果.另一问题是Dirichlet研究的双曲线下的整点个数问题.记D(R)表示在第一象限中双曲线xy=R与两个坐标轴所围成区域内的整点个数,则其中τ(n)为除数函数.因此这一问题又称为除数问题.Dirichlet首先证明了之后,上式中的余项不断被改进([11,13,17,18,21]).1985年,Kolesnik[14]将上式的余项改进至O(R139/429+ε).目前最好的结果是由Huxley[10]得到的,他将(2)式余项改进至O(R131/416+ε).此外,陈景润[2]和Vinogradov[20]分别研究了三维球面内满足n12+n22+n32≤x的整点个数问题,证明了后来Chamizo[1]和Heath-Brown[6]分别对上述结果的余项做出了改进.之后,许多数学家还研究了若干个整数的齐次方和表素数的问题.记2012年,郭汝庭和翟文广[4]对上式中s=3和k=2的情况进行了研究,证明了对任意取定的常数A>0,有其中B是某一个确定的常数.2014年,胡立群[7]还考虑了(3)式中s=4和k=2的情况,证明了对于任意取定的常数A>0,有其中C为某一个确定的常数.本文对更一般的s和k研究了(3)式并得到了如下结果.定理1 设Rs,k(x)如(3)式定义,那么当k≥2,k为偶数且s≥k2+k+1时,有其中C>0是一个任意取定的常数,并且更多还原

【Abstract】 The problem of lattice points is important in number theory. Firstly, Gauss and Dirichlet considered the problem of lattice points and raised two well-known problems, which are "The Gauss’s Circle Problem" and "The Dirichlet Divisor Problem". Denote by C(R) the number of lattice points in the circle x2+y2≤ R. Gauss proved that Afterwards, many researchers ([8,9,13,14,16,21]) investigated this problem and improved the above error term. In 1993, Huxley [9] improved the above error term to O(R23/73+ε). Later Huxley [10] refined the error term in (1) to O(R131/416+ε), which is the best result at present. Another problem is the number of lattice points under the hyperbola considered by Dirichlet. Denote by D(R) the number of lattice points between the two coordinate axes and the hyperbola xy= R in the first quartile. Then in which т(n) is the divisor function. Therefore, this problem called the divisor problem. Firstly Dirichlet proved that Later, the above error term was improved by many researchers ([11,13,17,18,21]). In 1985, Kolesnik [14] improved the above error term to O(R139、429+ε). Now the best result was obtained by Huxley [10], who refined the error term to O(R131/416+ε). In addition, Chen [2] and Vinogradov [20] studied the number of lattice points in the 3-dimensional ball n12+n22+n32≤ x independently. They proved that Later, Chamizo [1] and Heath-Brown [6] gave some improvements to the error term respectively.Moreover, many researchers are interested in the representation of primes, by sums of several homogeneous power integers. Let In 2012, Guo and Zhai [4] considered the case of s= 3 and k= 2 in (3), and proved that for any fixed positive constant A> 0, where B is a certain constant. Hu [7] studied the case of s= 4 and k= 2 in (3) in 2014 and proved that for any fixed positive constant A> 0, where C is a certain constant.In this paper, we will investigate (3) for general case of s and k and obtain the following result. Theorem 1 Let R(x) be defined as in (3). Then for k≥ 2, k even and s> k2+k+1, we have where C>0 is any fixed constant and更多还原