On representation of integers by sums of a cube and three cubes of primes
On representation of integers by sums of a cube and three cubes of primes
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DOI：
10.1307/mmj/1133894166
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发表时间：
2005-12
影响因子：
0.9
通讯作者：
Xiumin Ren;K. Tsang
中科院分区：
文献类型：
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作者：
Xiumin Ren;K. Tsang
我们考虑正整数n的表达式为一个立方与三个素数立方之和，即n = m + p2 + p33 + p34，（1.1）其中m为正整数，pj为素数. 1949年，Roth [6]证明了几乎所有的正整数n都可以写成（1.1）。确切地说，让E（N）表示N以上不能写成（1.1）的正整数的个数，那么罗斯定理实际上是说E（N）？N log−A N对于任意A > 0。这个结果可以被看作是猜想的一个近似，即所有满足某些必要的同余条件的足够大的整数都是四个素数的立方之和。众所周知，近似的质量在E（N）的上界中指示。Roth定理最近被Ren [3]改进为E（N）？N169/170，以及Ren和Tsang [4]到E（N）？N1271/1296+e.这些改进是通过新的方法来扩大所使用的圆方法中的主弧而获得的。这一点，见例[3]、[4]、[1]。本文在文献[4]中主弧估计的基础上，对次弧的处理采用了一些新的思想，并证明了如下结论。
We consider the expression of positive integers n as the sum of a cube and three cubes of primes, that is n = m + p2 + p 3 3 + p 3 4, (1.1) where m is a positive integer and pj are primes. In 1949, Roth [6] proved that almost all positive integers n can be written as (1.1). Precisely, let E(N) denote the number of positive integers up N which cannot be written as (1.1), then Roth’s theorem actually states that E(N) ? N log−A N for arbitrary A > 0. This result can be viewed as an approximation to the conjecture that all sufficiently large integers satisfying some necessary congruence conditions are the sum of four cubes of primes. As is well known that the quality of the approximation is indicated in the upper bound of E(N). Recently, Roth’s theorem has been improved by Ren [3] to E(N) ? N169/170, and by Ren and Tsang [4] to E(N) ? N1271/1296+e. These improvements were obtained via new approaches to enlarge major arcs in the circle method used. For this, see for example [3], [4], [1]. In this paper, based on the major arcs estimate in [4], we use some new ideas to handle the minor arcs and prove the following.