本文選題：華林-哥德巴赫問題 + Hardy-Littlewood方法��； 參考：《山東大學(xué)》2016年博士論文

【摘要】：1742年,哥德巴赫在與歐拉的兩封通信中,提出了著名的哥德巴赫猜想,具體可以表述為：(1)任何一個(gè)不小于6的偶數(shù),都可以表示成兩個(gè)奇素?cái)?shù)之和；(2)任何一個(gè)不小于9的奇數(shù),都可以表示成三個(gè)奇素?cái)?shù)之和.其中(1)被稱為偶數(shù)哥德巴赫猜想,(2)被稱為奇數(shù)哥德巴赫猜想.1937年,Vinogradov[64]基本解決了奇數(shù)哥德巴赫猜想.他借助Hardy-Littlewood方法并結(jié)合素變量三角和的估計(jì)證明了每一個(gè)充分大的奇數(shù)都可以表示成三個(gè)奇素?cái)?shù)之和.2013年,奇數(shù)哥德巴赫猜想被Helfgott[22,23]徹底解決.作為哥德巴赫問題的非線性推廣,華林-哥德巴赫問題也是數(shù)學(xué)家關(guān)注的焦點(diǎn)問題之一.這一類問題主要研究滿足某些同余條件的充分大的正整數(shù)N表為素?cái)?shù)方冪的可能性,即研究方程N(yùn)=p1k+p2k+…+psk (0.1)的可解性,其中p1,...,ps是素?cái)?shù).設(shè)H(k)表示使得對于所有滿足某些同余條件的充分大的N,方程(0.1)有解時(shí)s的最小值.對于固定的k,我們關(guān)心H(k)的上界.1938年,華羅庚[26]在對華林-哥德巴赫問題的研究中,以Hardy-Littlewood方法作為基本工具,結(jié)合Vinogradov關(guān)于素變量三角和的估計(jì)得到了深刻的結(jié)果.他證明了H(k)≤2k+1對于所有k≥1都成立.當(dāng)k≤3時(shí),這一結(jié)果仍然是迄今為止最好的.對于4≤k≤7的情況,H(k)的上界得到了很大程度的改進(jìn).到目前為止最好的結(jié)果為：H(4)≤13(趙立璐[72])；H(5)≤21(Kawada和Wooley[32]);H(6)≤32(趙立璐[72])；H(7)≤45(Kumchev和Wooley [36]).當(dāng)k≥8時(shí),目前最好的結(jié)果是由Kumchev和Wooley得到的,具體可參見文[36].這一系列結(jié)果,大部分是在應(yīng)用Hardy-Littlewood方法的基礎(chǔ)上得到的.根據(jù)Hardy在文[14]中的說法,Hardy-Littlewood方法應(yīng)該可以解決堆壘數(shù)論中包括哥德巴赫猜想、華林問題等在內(nèi)的許多經(jīng)典問題.雖然偶數(shù)哥德巴赫猜想至今尚未解決,但就堆壘數(shù)論問題的研究和發(fā)展來看,Hardy-Littlewood方法的確是研究堆壘數(shù)論問題的強(qiáng)有力工具之一本文我們將應(yīng)用Hardy-Littlewood方法研究幾類堆壘數(shù)論問題.考慮的第一個(gè)問題是幾乎相等的混合方次華林-哥德巴赫問題.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?cái)?shù).他證明了幾乎所有的偶數(shù)n都可以表示成(0.2)的形式.設(shè)E(N)表示不超過N且不能寫成(0.2)式的偶數(shù)n的個(gè)數(shù).在[51]中,Prachar證明了E(N)N(log N)-30/47+ε.后來Bauer[1,2],任秀敏和曾啟文[57,58]等都對E(N)的上界做了進(jìn)一步的研究.目前最好的結(jié)果是趙立璐在文[73]中得到的,他證明了E(N)N15/16+ε.在本文中,我們研究(0.2)式在素變量幾乎相等時(shí)的情形.具體來說,該問題是研究方程的可解性,其中U=N1-θ+ε,N是一個(gè)充分大的數(shù).令E(N,U)表示不能寫成(0.3)且滿足N-4U≤n≤N+4U的偶數(shù)n的個(gè)數(shù).在這一問題中,我們希望對盡可能大的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]中首先對這一問題進(jìn)行了研究并證明了(0.4)對于θ=1/264成立.本文也對方程(0.3)進(jìn)行了研究并改進(jìn)了[37]中的結(jié)果.我們的主要定理如下：定理1 在上述記號下,(0.4)式對θ=4/325成立.繼文[51]之后,Prachar在另一篇文章[52]中證明了任意充分大的奇數(shù)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)式在素變量幾乎相等時(shí)的情形,即研究了方程的可解性,其中U=N1-δ+ε.他們證明了當(dāng)θ=1/264時(shí)(0.6)式可解.在本文中,我們利用定理1,證明了下面這個(gè)結(jié)果.定理2 對于任意充分大的奇數(shù)N和U=N-14/325+ε,素變量方程(0.6)可解.本文考慮的第二個(gè)問題是與除數(shù)函數(shù)有關(guān)的一類均值估計(jì)問題.記d(n)為除數(shù)函數(shù),k是一個(gè)正整數(shù).對于X1,考慮關(guān)于除數(shù)函數(shù)的如下形式的均值：對于T(k,s；X)的估計(jì),以前的工作主要集中在k=2的情形.最早研究這個(gè)問題的是Gafurov,他在[10,11]中研究了當(dāng)s=2時(shí)的情形,證明了T(2,2;X)=A1X2logX+A2x2+O(X5/3 log9X),其中A1,A2是常數(shù).上式的余項(xiàng)被余剛在文[69]中改進(jìn)至O(X3/2+ε).2000年,C.Calder6n和M.J.de Velasco[6]研究了s=3時(shí)的情形.他們證明了后來這一結(jié)果被郭汝庭和翟文廣[13]改進(jìn)至其中Gi,Ij(i,j=1,2)是常數(shù).2014年,趙立璐[71]將上式余項(xiàng)改進(jìn)為O(X2 log7 X).此外,胡立群[24]證明了當(dāng)s=4時(shí),丁(2,4；X)有如下漸進(jìn)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)是常數(shù).隨后,胡立群和劉華鋒在文[251中,將上式余項(xiàng)進(jìn)一步改進(jìn)為O(X3 log 7 X).本文我們給出了當(dāng)k≥2時(shí)T(k,s;X)的漸進(jìn)公式.當(dāng)k=2時(shí),我們的主要結(jié)果如下：定理3 令T(k,s;X)如(0.7)式所定義且k=2.那么當(dāng)S≥3時(shí),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是任意給定的常數(shù),Gi,s和Ij,s(i,j=1,2)由(1.16)式定義.這里Ci,s(i=1,2)是該問題的奇異級數(shù),它是絕對收斂的且滿足Gi,s》1.注意到,當(dāng)s=3時(shí),定理3的結(jié)果與趙立璐在文[71]中的結(jié)果一致.當(dāng)s=4時(shí),定理3中的余項(xiàng)為O(X5/2 log X),此結(jié)果改進(jìn)了胡立群和劉華鋒在文[25]中的結(jié)果.對于k≥3,我們有下面的定理：定理4 設(shè)T(k,s;X)由(0.7)式定義且k≥3.那么對于smin{2k-1,k2+k-2},我們有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),其中這里Gi,k,s和Ij,k,s(i,j=1,2)由(1.14)式和(1.15)式定義.奇異級數(shù)Gi,k,s(i=1,2)是絕對收斂的且滿足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).
【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O156

【相似文獻(xiàn)】

相關(guān)期刊論文 前6條

1 陳景潤,王天澤;關(guān)于哥德巴赫問題[J];數(shù)學(xué)學(xué)報(bào);1989年05期

2 單土尊,闞家海;哥德巴赫問題的一個(gè)推廣[J];中國科學(xué)技術(shù)大學(xué)學(xué)報(bào);1997年02期

3 孟憲萌;;素變量具有特殊形式的哥德巴赫問題[J];數(shù)學(xué)學(xué)報(bào);2007年02期

4 岳玉靜;一個(gè)混合冪型的華林-哥德巴赫問題[J];山東大學(xué)學(xué)報(bào)(理學(xué)版);2005年02期

5 陳景潤,王天澤;關(guān)于哥德巴赫問題的一點(diǎn)注記[J];數(shù)學(xué)學(xué)報(bào);1991年01期

6 ;[J];;年期

相關(guān)博士學(xué)位論文 前4條

1 藥艷君;幾乎相等的華林—哥德巴赫問題及相關(guān)問題[D];山東大學(xué);2015年

2 張萌;Hardy-Littlewood方法的若干應(yīng)用[D];山東大學(xué);2016年

3 徐云飛;小區(qū)間內(nèi)的華林—哥德巴赫問題[D];山東大學(xué);2006年

4 趙峰;小區(qū)間上的三次華林—哥德巴赫問題[D];山東大學(xué);2010年

相關(guān)碩士學(xué)位論文 前4條

1 王勿匆;小區(qū)間上的華林—哥德巴赫問題[D];山東大學(xué);2011年

2 許澤;表自然數(shù)為一個(gè)素?cái)?shù)與五個(gè)素?cái)?shù)立方之和的小區(qū)間問題[D];山東大學(xué);2014年

3 王英男;用Green-Tao的觀點(diǎn)看奇數(shù)哥德巴赫問題[D];山東大學(xué);2008年

4 劉亮;表自然數(shù)為一個(gè)素?cái)?shù)和三個(gè)素?cái)?shù)平方之和的小區(qū)間問題[D];山東大學(xué);2009年

，

本文編號：2085148

boshi