發(fā)布時(shí)間：2018-05-10 19:55

  本文選題：Waring-Goldbach問(wèn)題 + Harman篩法�。� 參考：《山東大學(xué)》2017年博士論文

【摘要】：令n是滿足某些局部同余條件的充分大的整數(shù),κ是一個(gè)正整數(shù).Waring-Goldbach問(wèn)題主要研究將整數(shù)n表示為素?cái)?shù)的方冪之和,即n = p1κ + p2κ + …+psκ,(0.1)其中p1…,ps表示素?cái)?shù).如果取κ= 1,s = 2,則上面的問(wèn)題就是至今尚未得到解決的Goldbach猜想(偶數(shù)Goldbach猜想),也就可以認(rèn)為Waring-Goldbach問(wèn)題是Goldbach問(wèn)題的非線性推廣.關(guān)于Waring-Goldbach問(wèn)題的線性情形,Vinogradov[44]在1937年證明了當(dāng)s ≥ 3時(shí),對(duì)于每一個(gè)充分大的奇數(shù)n,方程(0.1)都存在奇素?cái)?shù)解,這被稱作著名的三素?cái)?shù)定理.2013年,Helfgott[9,10]證明了當(dāng)s≥3時(shí),對(duì)所有大于等于9的奇數(shù)n,方程(0.1)都存在奇素?cái)?shù)解,完全解決了奇數(shù)Goldbach猜想.關(guān)于Waring-Goldbach問(wèn)題的非線性情形,1938年華羅庚在[11]首先證明了當(dāng)s≥2κ+ 1時(shí),方程(0.1)對(duì)所有的κ≥1都存在素?cái)?shù)解,并在[12]中進(jìn)行了系統(tǒng)地總結(jié).該結(jié)果在κ ≤ 3時(shí)仍然是最好的結(jié)果.對(duì)于≥ 4的情形,許多學(xué)者改進(jìn)了這一結(jié)果(參見[15,16,18,19,39,40,42,49]).數(shù)論領(lǐng)域另外一個(gè)非常有意義的問(wèn)題是變量幾乎相等的Waring-Goldbach 問(wèn)題.接下來(lái),我們 對(duì)這一 問(wèn)題進(jìn) 行詳細(xì) 地說(shuō)明.首先,我們令τ=τ(κ p)為滿足pτ|κ的最大的整數(shù),同時(shí)定義(?)(?)(?)其他.將整數(shù)n限制在同余類(?)中,我們來(lái)研究方程(0.1)解的情況.給定一個(gè)充分大的整數(shù)n ∈Hk,s,變量幾乎相等的Waring-Goldbach問(wèn)題主要研究方程(0.1)是否存在滿足關(guān)于變量幾乎相等的Waring-Goldbach問(wèn)題,對(duì)于k= 2,s = 5時(shí)的情形有很多的結(jié)果(參見[2,3,4,17,24,25,26,27,29,30,35]).特別地,1996年,劉建亞和展?jié)齕25]最先考慮這一問(wèn)題.2012年,Kumchev和李太玉[17]得到關(guān)于該問(wèn)題目前最好的結(jié)果:對(duì)任意固定的θ8/9,方程(0.1)存在滿足(0.2)的素?cái)?shù)解,此時(shí)(0.2)中的H = nθ/2.同時(shí)他們最先得到變量個(gè)數(shù)多于五個(gè)的幾乎相等的素?cái)?shù)的平方和的結(jié)果,其中多余的變量是用來(lái)減小可允許的H的大小.記H= nθ/k.令θk,s表示方程(0.1)對(duì)充分大的,n∈Hk,s,存在滿足(0.2)的素?cái)?shù)解的θ的最小值.Kumchev和李太玉[17]證明了當(dāng)s ≥ 17時(shí),θ2,s ≤ 19/24.2014年,魏斌和Wooley[45]將s的下界改進(jìn)到s ≥ 7;同時(shí)他們還得到了更高次的結(jié)果:當(dāng)s＞2k(k-1)時(shí),2016年,黃炳榮[13]證明了對(duì)所有的k≥ 3和s2k(k-1),均有θk,s ≤ 19/24,進(jìn)一步改進(jìn)了魏斌和Wooley[45]的結(jié)果.本文主要利用Harman篩法突破主區(qū)間對(duì)θ的限制,相比之前擴(kuò)大了θ的取值范圍,在一定程度上可以說(shuō)做到了目前最好的結(jié)果.同時(shí)我們也利用了Bourgain,Demeter和Guth[5]的最新結(jié)果,改進(jìn)了當(dāng)k≥4時(shí)s的下界.我們進(jìn)一步改進(jìn)了黃炳榮[13]的結(jié)果.本文的主要結(jié)果如下:定理1 令k ≥ 2,s≥k2+k+1和θ31/40.則對(duì)于每一個(gè)充分大的整數(shù)n ∈ Hk,s,方程(0.1)存在滿足(0.2)的素?cái)?shù)解p1,…,,Ps.Waring-Goldbach問(wèn)題的例外集問(wèn)題也是數(shù)論領(lǐng)域的一個(gè)重要問(wèn)題,讀者可以參考文章[17,28,31,38]來(lái)詳細(xì)地了解關(guān)于這一問(wèn)題的發(fā)展過(guò)程.在同一篇文章中,魏斌和Wooley[45]還得到了關(guān)于方程(0.1)對(duì)"幾乎所有"的n的可解性和關(guān)于六個(gè)幾乎相等的素?cái)?shù)平方和的例外集兩個(gè)問(wèn)題的結(jié)果.黃炳榮[13]改進(jìn)了前一個(gè)問(wèn)題的結(jié)果.不難看出,根據(jù)定理1的證明和文章[45,§9]中的方法,我們可以進(jìn)一步改進(jìn)上述兩個(gè)問(wèn)題的結(jié)果.我們有下面兩個(gè)結(jié)果:定理2 令κ≥2,s ＞κ(κ+ 1)/2,θ31/40和N→ ∞.則存在一個(gè)固定的δ0,使得除去O(N1-δ)個(gè)以外,幾乎對(duì)所有的整數(shù)n≤ N且n ∈Hκ,s 方程(0.1)存在有滿足(0.2)的素?cái)?shù)解p1,…,ps(當(dāng)κ=3,s= 7時(shí),9(?)n).令E6(N;H)表示滿足以下條件的整數(shù)n的個(gè)數(shù):a.|n-N|≤ HN1/2,b.n = 6(mod 24),c.取= 2,s = 6,方程(0.1)不存在滿足(0.2)的素?cái)?shù)解內(nèi),….,Ps.定理3 令θ31/40和N →∞.則存在一個(gè)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.

【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O156

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