Representation Of Primes By Sums Of K-th Powers Of Integers

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...