关于变量几乎相等的Waring-Goldbach问题

发布时间：2018-05-10 19:55

本文选题：Waring-Goldbach问题 + Harman筛法　；参考：《山东大学》2017年博士论文

【摘要】：令n是满足某些局部同余条件的充分大的整数,κ是一个正整数.Waring-Goldbach问题主要研究将整数n表示为素数的方幂之和,即n = p1κ + p2κ + …+psκ,(0.1)其中p1…,ps表示素数.如果取κ= 1,s = 2,则上面的问题就是至今尚未得到解决的Goldbach猜想(偶数Goldbach猜想),也就可以认为Waring-Goldbach问题是Goldbach问题的非线性推广.关于Waring-Goldbach问题的线性情形,Vinogradov[44]在1937年证明了当s ≥ 3时,对于每一个充分大的奇数n,方程(0.1)都存在奇素数解,这被称作著名的三素数定理.2013年,Helfgott[9,10]证明了当s≥3时,对所有大于等于9的奇数n,方程(0.1)都存在奇素数解,完全解决了奇数Goldbach猜想.关于Waring-Goldbach问题的非线性情形,1938年华罗庚在[11]首先证明了当s≥2κ+ 1时,方程(0.1)对所有的κ≥1都存在素数解,并在[12]中进行了系统地总结.该结果在κ ≤ 3时仍然是最好的结果.对于≥ 4的情形,许多学者改进了这一结果(参见[15,16,18,19,39,40,42,49]).数论领域另外一个非常有意义的问题是变量几乎相等的Waring-Goldbach 问题.接下来,我们对这一问题进行详细地说明.首先,我们令τ=τ(κ p)为满足pτ|κ的最大的整数,同时定义(?)(?)(?)其他.将整数n限制在同余类(?)中,我们来研究方程(0.1)解的情况.给定一个充分大的整数n ∈Hk,s,变量几乎相等的Waring-Goldbach问题主要研究方程(0.1)是否存在满足关于变量几乎相等的Waring-Goldbach问题,对于k= 2,s = 5时的情形有很多的结果(参见[2,3,4,17,24,25,26,27,29,30,35]).特别地,1996年,刘建亚和展涛[25]最先考虑这一问题.2012年,Kumchev和李太玉[17]得到关于该问题目前最好的结果:对任意固定的θ8/9,方程(0.1)存在满足(0.2)的素数解,此时(0.2)中的H = nθ/2.同时他们最先得到变量个数多于五个的几乎相等的素数的平方和的结果,其中多余的变量是用来减小可允许的H的大小.记H= nθ/k.令θk,s表示方程(0.1)对充分大的,n∈Hk,s,存在满足(0.2)的素数解的θ的最小值.Kumchev和李太玉[17]证明了当s ≥ 17时,θ2,s ≤ 19/24.2014年,魏斌和Wooley[45]将s的下界改进到s ≥ 7;同时他们还得到了更高次的结果:当s＞2k(k-1)时,2016年,黄炳荣[13]证明了对所有的k≥ 3和s2k(k-1),均有θk,s ≤ 19/24,进一步改进了魏斌和Wooley[45]的结果.本文主要利用Harman筛法突破主区间对θ的限制,相比之前扩大了θ的取值范围,在一定程度上可以说做到了目前最好的结果.同时我们也利用了Bourgain,Demeter和Guth[5]的最新结果,改进了当k≥4时s的下界.我们进一步改进了黄炳荣[13]的结果.本文的主要结果如下:定理1 令k ≥ 2,s≥k2+k+1和θ31/40.则对于每一个充分大的整数n ∈ Hk,s,方程(0.1)存在满足(0.2)的素数解p1,…,,Ps.Waring-Goldbach问题的例外集问题也是数论领域的一个重要问题,读者可以参考文章[17,28,31,38]来详细地了解关于这一问题的发展过程.在同一篇文章中,魏斌和Wooley[45]还得到了关于方程(0.1)对"几乎所有"的n的可解性和关于六个几乎相等的素数平方和的例外集两个问题的结果.黄炳荣[13]改进了前一个问题的结果.不难看出,根据定理1的证明和文章[45,§9]中的方法,我们可以进一步改进上述两个问题的结果.我们有下面两个结果:定理2 令κ≥2,s ＞κ(κ+ 1)/2,θ31/40和N→ ∞.则存在一个固定的δ0,使得除去O(N1-δ)个以外,几乎对所有的整数n≤ N且n ∈Hκ,s 方程(0.1)存在有满足(0.2)的素数解p1,…,ps(当κ=3,s= 7时,9(?)n).令E6(N;H)表示满足以下条件的整数n的个数:a.|n-N|≤ HN1/2,b.n = 6(mod 24),c.取= 2,s = 6,方程(0.1)不存在满足(0.2)的素数解内,….,Ps.定理3 令θ31/40和N →∞.则存在一个固定的δ0使得E6(N;Nθ/2)N(1-θ)/2-δ.
[Abstract]:N is a sufficiently large integer to satisfy some local congruence conditions. Kappa is a positive integer.Waring-Goldbach problem that mainly studies the sum of the power of the integer n as the prime number, that is, n = P1 kappa + P2 kappa +... +ps kappa, (0.1) in which P1... PS is a prime number. If kappa = 1, s = 2, then the above problem is the Goldbach conjecture (even Goldbach conjecture) that has not been solved so far, and we can consider the Waring-Goldbach problem to be a nonlinear generalization of the Goldbach problem. On the linear case of Waring-Goldbach problem, Vinogradov[44] in 1937 proved that when s is equal to 3, for each of the Goldbach, Vinogradov[44] has been proved to be a problem. A sufficiently large odd number n, the equation (0.1) has an odd prime number solution, which is called the famous three prime number theorem.2013. Helfgott[9,10] proves that when s is more than 3, all the odd number n, which is greater than or equal to 9, has an odd prime number solution, which completely solves the odd number Goldbach conjecture. The nonlinear case about the Waring-Goldbach problem, the 1938 year's time. In [11], [11] first proved that when s > 2 kappa + 1, the equation (0.1) has a prime number solution for all kappa > 1 and has been systematically summarized in [12]. This result is still the best result when kappa < 3. For the case of more than 4, many scholars have improved this result (see [15,16,18,19,39,40,42,49]). Another very much in the field of number theory. A meaningful problem is a Waring-Goldbach problem with almost equal variables. Next, we give a detailed description of this problem. First, we make tau = tau (kappa P) to satisfy the largest integer of P [tau] kappa, and define (?) (?) other. We limit the integer n to the congruent class (?), we study the equation (0.1) solution. The large integer n Hk, s, the Waring-Goldbach problem with almost equal variables is the main study of whether the equation (0.1) exists to satisfy the Waring-Goldbach problem of almost equal variables. There are many results for the case of k= 2, s = 5 (see [see 2,3,4,17,24,25,26,27,29,30,35]). In particular, in 1996, Liu Jianya and Exhibition Tao [25] were first considered. This problem.2012, Kumchev and Li Tai Yu [17] get the best result of the problem at present: for any fixed theta 8/9, the equation (0.1) has a prime solution that satisfies (0.2), and the H = n theta /2. in (0.2) at the same time they first get the result that the variable number is more than the square sum of the equal prime number more than five, of which the superfluous variable is To reduce the size of the permissible H. H= n theta /k. orders theta K, s to express the equation (0.1) to the sufficient large, n Hk, s, the minimum value of the theta of the prime solution of (0.2),.Kumchev and Li Tai Yu [17] proves that when s is equal to 17, theta 2, the lower bound is improved to more than 7; and they also get higher order. Results: when s > 2K (k-1), in 2016, Huang Ming [13] proved that all k > 3 and S2K (k-1) all have theta K, s < 19/24, further improved the results of Wei Bin and Wooley[45]. This paper mainly uses the Harman sieve method to break through the limit of the main interval to theta. Compared with before, the range of theta is expanded, to a certain extent, it is possible to do the present. At the same time, we also use the latest results of Bourgain, Demeter and Guth[5] to improve the lower bounds of s when k > 4. We further improved the results of Huang Ping Rong's [13]. The main results of this article are as follows: Theorem 1 k > 2, s > k2+k+1 and theta 31/40. for each full n integer n Hk, governing, equation (0.1) existence (0.1). 0.2) the prime number solution P1,... The exception set problem of the Ps.Waring-Goldbach problem is also an important problem in the field of number theory. Readers can refer to the article [17,28,31,38] to understand the development of this problem in detail. In the same article, Wei Bin and Wooley[45] also obtained the solvability of the equation (0.1) for "almost all" of N and about six almost. The results of two problems with the exception set of the sum of the sum of prime square sum. Huang [13] improved the result of the previous problem. It is not difficult to see that according to the proof of Theorem 1 and the article [45, we can further improve the results of the above two problems. We have the following two results: theorem 2 order kappa > 2, s > kappa (kappa + 1) /2, theta 31/40 and N There is a fixed delta 0, so that except for O (N1- delta), almost all integers n < N and N H kappa, s equation (0.1) has a prime solution of P1, which satisfies (0.2),... PS (when kappa =3, s= 7, 9 (?) n). Order E6 (N; H) to represent the number of integers n that satisfies the following conditions: a.|n-N| < HN1/2, B.N = 6 (MOD 24), = 2, 6, equation (0.1) does not exist in a prime solution (0.2),... The Ps. Theorem 3 makes theta 31/40 and N > infinity. Then there is a fixed delta 0 so that E6 (N, N theta /2) N (1- theta) /2- Delta.

【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O156

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