Hardy-Littlewood方法的若干应用

发布时间：2018-06-30 05:38

本文选题：华林-哥德巴赫问题 + Hardy-Littlewood方法　；参考：《山东大学》2016年博士论文

【摘要】：1742年,哥德巴赫在与欧拉的两封通信中,提出了著名的哥德巴赫猜想,具体可以表述为：(1)任何一个不小于6的偶数,都可以表示成两个奇素数之和；(2)任何一个不小于9的奇数,都可以表示成三个奇素数之和.其中(1)被称为偶数哥德巴赫猜想,(2)被称为奇数哥德巴赫猜想.1937年,Vinogradov[64]基本解决了奇数哥德巴赫猜想.他借助Hardy-Littlewood方法并结合素变量三角和的估计证明了每一个充分大的奇数都可以表示成三个奇素数之和.2013年,奇数哥德巴赫猜想被Helfgott[22,23]彻底解决.作为哥德巴赫问题的非线性推广,华林-哥德巴赫问题也是数学家关注的焦点问题之一.这一类问题主要研究满足某些同余条件的充分大的正整数N表为素数方幂的可能性,即研究方程N=p1k+p2k+…+psk (0.1)的可解性,其中p1,...,ps是素数.设H(k)表示使得对于所有满足某些同余条件的充分大的N,方程(0.1)有解时s的最小值.对于固定的k,我们关心H(k)的上界.1938年,华罗庚[26]在对华林-哥德巴赫问题的研究中,以Hardy-Littlewood方法作为基本工具,结合Vinogradov关于素变量三角和的估计得到了深刻的结果.他证明了H(k)≤2k+1对于所有k≥1都成立.当k≤3时,这一结果仍然是迄今为止最好的.对于4≤k≤7的情况,H(k)的上界得到了很大程度的改进.到目前为止最好的结果为：H(4)≤13(赵立璐[72])；H(5)≤21(Kawada和Wooley[32]);H(6)≤32(赵立璐[72])；H(7)≤45(Kumchev和Wooley [36]).当k≥8时,目前最好的结果是由Kumchev和Wooley得到的,具体可参见文[36].这一系列结果,大部分是在应用Hardy-Littlewood方法的基础上得到的.根据Hardy在文[14]中的说法,Hardy-Littlewood方法应该可以解决堆垒数论中包括哥德巴赫猜想、华林问题等在内的许多经典问题.虽然偶数哥德巴赫猜想至今尚未解决,但就堆垒数论问题的研究和发展来看,Hardy-Littlewood方法的确是研究堆垒数论问题的强有力工具之一本文我们将应用Hardy-Littlewood方法研究几类堆垒数论问题.考虑的第一个问题是几乎相等的混合方次华林-哥德巴赫问题.1953年,Prachar在文[51]中首先研究了下面方程的可解性n=p22+p33+p44+p55,(0.2)其中pi(2≤if≤5)是素数.他证明了几乎所有的偶数n都可以表示成(0.2)的形式.设E(N)表示不超过N且不能写成(0.2)式的偶数n的个数.在[51]中,Prachar证明了E(N)N(log N)-30/47+ε.后来Bauer[1,2],任秀敏和曾启文[57,58]等都对E(N)的上界做了进一步的研究.目前最好的结果是赵立璐在文[73]中得到的,他证明了E(N)N15/16+ε.在本文中,我们研究(0.2)式在素变量几乎相等时的情形.具体来说,该问题是研究方程的可解性,其中U=N1-θ+ε,N是一个充分大的数.令E(N,U)表示不能写成(0.3)且满足N-4U≤n≤N+4U的偶数n的个数.在这一问题中,我们希望对尽可能大的θ∈(0,1),有E(N,U)《U1-ε 其中 U=N1-θ+ε.(0.4)2012年,李太玉和唐恒才在[37]中首先对这一问题进行了研究并证明了(0.4)对于θ=1/264成立.本文也对方程(0.3)进行了研究并改进了[37]中的结果.我们的主要定理如下：定理1 在上述记号下,(0.4)式对θ=4/325成立.继文[51]之后,Prachar在另一篇文章[52]中证明了任意充分大的奇数N都可以表示成N=p1+p22+p33+p44+55. (0.5)在[37]中,李太玉和唐恒才研究了(0.5)式在素变量几乎相等时的情形,即研究了方程的可解性,其中U=N1-δ+ε.他们证明了当θ=1/264时(0.6)式可解.在本文中,我们利用定理1,证明了下面这个结果.定理2 对于任意充分大的奇数N和U=N-14/325+ε,素变量方程(0.6)可解.本文考虑的第二个问题是与除数函数有关的一类均值估计问题.记d(n)为除数函数,k是一个正整数.对于X1,考虑关于除数函数的如下形式的均值：对于T(k,s；X)的估计,以前的工作主要集中在k=2的情形.最早研究这个问题的是Gafurov,他在[10,11]中研究了当s=2时的情形,证明了T(2,2;X)=A1X2logX+A2x2+O(X5/3 log9X),其中A1,A2是常数.上式的余项被余刚在文[69]中改进至O(X3/2+ε).2000年,C.Calder6n和M.J.de Velasco[6]研究了s=3时的情形.他们证明了后来这一结果被郭汝庭和翟文广[13]改进至其中Gi,Ij(i,j=1,2)是常数.2014年,赵立璐[71]将上式余项改进为O(X2 log7 X).此外,胡立群[24]证明了当s=4时,丁(2,4；X)有如下渐进公式：T(2,4;X)=2C'1I'1X4 log X+(C'1I'2+C'2I'1)X4+O(X7/2+ε),其中C'i,I'j(i,j=1,2)是常数.随后,胡立群和刘华锋在文[251中,将上式余项进一步改进为O(X3 log 7 X).本文我们给出了当k≥2时T(k,s;X)的渐进公式.当k=2时,我们的主要结果如下：定理3 令T(k,s;X)如(0.7)式所定义且k=2.那么当S≥3时,有T(2,s;X)=2C1,sT1,sXs logX+(C1,sI2,s+C2,sI1,s)Xs+Os(Xs+1/2log s+4X+Xs-2 log X),其中ε0是任意给定的常数,Gi,s和Ij,s(i,j=1,2)由(1.16)式定义.这里Ci,s(i=1,2)是该问题的奇异级数,它是绝对收敛的且满足Gi,s》1.注意到,当s=3时,定理3的结果与赵立璐在文[71]中的结果一致.当s=4时,定理3中的余项为O(X5/2 log X),此结果改进了胡立群和刘华锋在文[25]中的结果.对于k≥3,我们有下面的定理：定理4 设T(k,s;X)由(0.7)式定义且k≥3.那么对于smin{2k-1,k2+k-2},我们有T(k,s;X)=kC1,k,sII,k,sXs log X+(C1,k,s,2,k,s+C2,k,sI1,k,s)Xs+O(Xs-θ+ε),其中这里Gi,k,s和Ij,k,s(i,j=1,2)由(1.14)式和(1.15)式定义.奇异级数Gi,k,s(i=1,2)是绝对收敛的且满足Ci,k,s1.
[Abstract]:In 1742, in two correspondence with Euler, Goldbach proposed the famous Goldbach conjecture, which can be expressed as: (1) any even number of no less than 6 can be expressed as the sum of two odd prime numbers; (2) any odd number of no less than 9 can be shown as the sum of three odd prime numbers. (1) is called the even number Goldba. The Herr conjecture, (2) known as the odd number Goldbach conjecture.1937, Vinogradov[64] basically solved the odd number Goldbach's conjecture. He proved that every sufficient large odd number can be represented as three odd prime and.2013 years with the Hardy-Littlewood method and the estimation of the prime variable triangle, and the odd Goldbach conjecture is Helfgott[22, 23] is a complete solution. As a nonlinear generalization of Goldbach problem, Hualin Goldbach problem is also one of the focus problems of mathematicians. This kind of problem mainly studies the possibility that the large positive integer N table which satisfies some congruence conditions is the power of prime square, that is, the study of the equation N= p1k+p2k+... The solvability of +psk (0.1), in which P1,..., and PS are prime numbers. Set H (k) to express the minimum value of s when there is a sufficient N for all the congruence conditions. For a fixed k, we care about the upper bound of H (k), and Hua [26] is the basic of the study of the Hua Lin Goldbach problem. The tool, combined with Vinogradov's estimation of the sum of the prime variable triangles, is a profound result. He proved that H (k) < 2k+1 is good for all k > 1. When k is less than 3, this result is still the best so far. For the case of 4 less than k < 7, the upper bound of H (k) has been greatly improved. So far, the best result is: H (4) < 1. 3 (Zhao Lilu [72]); H (5) < 21 (Kawada and Wooley[32]); H (6) < 32 (Zhao Li Lu [72]); H (7) < 45 (Kumchev and Wooley [36]). When k > 8, the present best result is obtained by Kumchev and other results, most of which are obtained on the basis of the application method. In [14], the Hardy-Littlewood method should be able to solve many classical problems, including Goldbach's conjecture and Hualin problem in the stack base number theory. Even though even Goldbach's conjecture has not been solved yet, the study and development of the problem of the number theory of heap base, the Hardy-Littlewood method is indeed a strong study of the problem of the number theory of heap base. One of the powerful tools in this paper we will use the Hardy-Littlewood method to study the problem of a number of heap number theory. The first problem to be considered is the almost equal mixed square Hualin Goldbach problem.1953. Prachar first studied the solvability n=p22+p33+p44+ p55 of the following equation in [51], (0.2) PI (2 < if < 5) is a prime number. It is clear that almost all even numbered n can be represented as (0.2) form. Set E (N) to express no more than N and can not be written as a number of even n of (0.2). In [51], Prachar proves E (N) N (log N). In this paper, he proved E (N) N15/16+ epsilon. In this article, we study the case of (0.2) in the case that the prime variable is almost equal. Specifically, the problem is to study the solvability of the equation, in which U=N1- theta + epsilon, N is a sufficiently large number. The E (N, U) representation can not be written as (0.3) and satisfies the number of even numbers of N-4U < < < n > N+4U. In one problem, we want to be as large as possible (0,1), E (N, U) 1. notes that when s=3, the result of Theorem 3 coincide with the result of Zhao Li Lu in the article [71]. When s=4, the remainder in Theorem 3 is O (X5/2 log X), and this result improves the results of Hu Liqun and Liu Huafeng in the article [25]. K > 3., then smin{2k-1, k2+k-2}, we have T (k, s; X) =kC1, K, sII, K, which are defined by (1.14) and (1.15).
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O156

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