摘要
华林-哥德巴赫问题研究的是将满足一定同余条件的正整数用素数幂表示的问题.著名的哥德巴赫猜想和三素数定理就是华林-哥德巴赫问题线性情况下的个例.众所周知,解决华林-哥德巴赫问题的一般性的方法是Hardy和Littlewood的圆法结合Vinogradov素数上的指数和的估计.1965年以前的结果已经总结于华罗庚《堆垒素数论》一书中.自那以后,特别是近些年,圆法,筛法和指数和的新思想不断地被运用到华林-哥德巴赫问题中,并得到了很多显著的成果.
小区间的华林-哥德巴赫问题也吸引了许多数学工作者,他们在这一方面得到了许多的结果.其中,以几乎相等的素变量哥德巴赫-维诺格拉多夫定理最为著名.
华林-哥德巴赫问题主要分线性和非线性两种情形,不同于线形情形,用圆法来研究非线性情形时需克服更大的困难.具体来讲,在应用圆法处理非线性的情形的时候需要处理更大的主区间.为了克服这个困难,刘建亚教授和展涛教授[11]首先在广义黎曼猜想下研究了小区间上二次非线性情形下的华林-哥德巴赫问题.更确切的说,他们在广义黎曼猜想下证明了:每个模24同余于5的大整数N可以表示成5个几乎相等的素数的平方之和,即有解,其中,U=(?).
Bauer在文章[2]中无条件的证明了(0.1)式对于U=N~(1/2-δ)成立,其中δ≥0.δ的值依赖于Deuring-Heilbronn现象的常数值,是不可算的.
1998年,刘建亚教授和展涛教授[11]在研究华林-哥德巴赫问题时找到了处理扩大主区间的新方法,这种方法的引入使得可能存在的Siegel零点对定理不再有影响,所以,Deuring-Heilbronn现象可以被避免了.用这种方法,他们无条件的证明了(0.1)对U = N~(1/2-1/50+ε)是成立的.随着这种方法的不断成熟和改进,许多无条件的结果陆续得出。Bauer将上述问题的指数值由(?)改进到了(?),后来又被吕广世教授改进到了(?).现在,这一方法已经被成功的运用到许多素数的加性问题中.
在这篇文章里,我们将应用刘和展的扩大主区间的思想研究华罗庚先生表自然数为一个素数和三个素数平方之和的小区间问题.主要结论如下:
定理对于每个充分大的整数N(?)0(mod3),方程对于U=(?)有解.
The Waring-Goldbach problem seeks to represent positive integers satisfying necessary congruence conditions by powers of primes, the ternary and binary Goldbach problems are just liner examples of the Waring-Goldbach problem.
The circle method of Hardy and Littlewood in combination with the estimates of Vinogiadov for exponential sums over primes gives an affirmative answer to the general Waring-Goldbach problem, and the results before 1965 was summarized in Hua's book "Additive Theory on Prime Numbers". After that, especially in resent years, new ideas in the circle method, sieves, and exponential sums arc incorporated into the Waring-Goldbach problem, and hence give remarkable advances.
Waring-Goldbach problems in short intervals have appealed to many authors and have been investigated, among which Goldbach-Vinogradov theorem with almost equal prime variables may be the most famous one .
Different from linear case, non-linear cases have to treat, the enlarged major arcs when applying the circle method. In order to avoid this difficulty, Liu and Zhan [11] first studied the quadratic case assuming the Generalized Riemann Hypothesis (GRE). More precisely, they showed that under GRE each large integer N≡5(mod24) can be written aswhere U=N~(1/2-δ+ε),δ=(?).
Later Bauer [2] unconditionally showed that the formula (0.1) holds true for U = N~(1/2-δ), whereδ≥0 and its exact value depends on the constants in the Deuring-Heilbronn phenomenon, and is not numerically determined.
In 1998 Liu and Zhan [11] found the new approach to treat the enlarged major arcs in which the possible existence of Siegel zero does not have special influence, and hence the During-Heibornn phenomenon can be avoided. Due to this approach, they unconditionally obtained that (0.1) is true for U = N~(1/2-1/50+ε). With the development of this approach, the exponent (?) has subsequently reduced to (?) by Baucr and then to (?) by Guang-Shi L(?).
In this paper we study one Waring-Goldbach problem in short intervals .
We establish the following result as a short interval version of Hua's theorem on the sum of a prime and three squares of primes (see [5]).
Theorem For each sufficiently large even integer not congruent to 0(mod 3), the equation in prime variableshas solution for U = (?),
引文
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