摘要
在加性数论中,人们经常研究将一个正整数表示成素数幂之和的可能性.华林-哥德巴赫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[1]就是华林-哥德巴赫问题线性情况下的特例.众所周知,华林-哥德巴赫问题的一般性方法是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.
另一方面,研究在适当限制条件下的对垒素数问题也吸引了大量的数学工作者.人们猜想所有充分大的满足一定同余条件的正整数是四个素数的立方和.但是,这么强的一个结果人们还不能够证明.这一方向最好的结论是由华[2]于1938年所证明的：
·每一个充分大的整数是9个素数的立方和；
·适合于集合(?)={n≥1：n≠0,±2(mod 9)}的整数是5个素数的立方和,即有：(0.1)
更精确的,记E(N)表示那些不能写为(0.1)式的整数的个数,其中n∈(?)且不超过N.那么华的第二个结论表明其中,A>0是任意的.
借助圆法和其他方法,我们可以对这一结果进行改进.其中,任[3]做出了如下的改进：
为了得到这样的结果,必须处理更大的主区间.在1998年,刘和展[4]找到了一种新的方法来扩大华林-哥德巴赫问题的主区间.正是这种技巧,研究者可以忽略Siegel零的影响,避免Deuring-Heilbronn现象,更好地研究堆垒素数问题.
在这篇文章里,将应用刘和展的扩大主区间的思想,研究这一问题的推广.在华[2]的文章里,每一个满足一定条件的充分大的奇整数是5个素数的立方和,我们将其中的一个素变数变为整变数,研究表一个充分大的整数为一个整数和四个素数立方和的整数分布问题,即若n∈(?),就有其中m1是一个正整数.
本文的结果如下：
定理若E(N)如上所述,则有
In the additive theory of prime numbers, one studies the representation of positive integers by powers of primes. The Waring-Goldbach problem seeks to represent positive integers satisfying necessary congruence conditions by powers of primes. The ternary and binary Goldbach problems[1] are just liner examples of the Waring-Goldbach prob-lem.
The circle method of Hardy and Littlewood in combination with the estimates of Vinogradov 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 Primes Numbers". After that, especially in resent yeas, new ideas in the circle method,sieves, and exponential sums are incorporated into the Waring-Goldbach problem, and hence give remarkable advance.
On the other hand, the additive theory of prime numbers with certain conditions appeals to many researchers to study. It is conjectured that all sufficiently large integers satisfying some necessary congruence conditions are sums of four cubes of primes. Such a strong result is out of reach at present. The best result in this direction is due to Hua[2] and dates back to 1938:
●All sufficiently large integers are sums of nine cubes of primes;
●Almost all integers n in the set n={n≥1:n(?)0,±2 (mod 9)} can be represented as sums of five cubes of primes, i.e. (0.1)
To be more precise, let E(N) denote the number of integers n∈n not exceeding N which cannot be written as the formula(0.1) mentioned above. Then Hua's second result actually states that where A> 0 is arbitrary.
With the help of circle method, we can get advanced result for this question. And Ren[3] gave the following result:
To get a result of this strength, we have to deal with rather large major arcs. in 1998, Liu and Zhan[4] found a new approach to treat the enlarged major arcs in the Waring-Glodbach problem, in which the possible existence of the Sicgel zero does not have a special influence, and hence the Deuring-Heilbronn phenomenon can be avoided.
In this paper, with the help of circle method, we study the generalization of the problem. Hua in [2] proved that every sufficiently large integer satisfying necessary conditions is the sums of five cubes of primes. We replace a prime with a integer, studying the representation of integers by sums of a cube and four cubes of primes, that is. if n∈n, then where m1 is a positive integer.
We shall prove the following theorem.
Theorem If E(N) is as above, then
引文
[1]Vinogradov, I. M., Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk SSSR,15(1937),291-294.
[2]L. K. Hua, Some results in the additive prime number theory, Quart. J. Math.,9(1938), 68-80.
[3]X. Ren, The Waring-Goldbach problem for cubes, Acta Arith.,94(2000),287-301.
[4]J. Y. Liu and T. Zhan, Sums of five almost equal prime squares, Science in China, Series A,7(41), (1998),710-722.
[5]K. F. Roth, On Waring's problem for cubes, Prov. London Math. Soc., (2)53(1951), 268-279.
[6]X. Ren, The exceptional set in Roth's theorem concering a cube and three cubes of primes, Quart. J. Math. (Oxford),52(2001),107-126.
[7]X. Ren and K. M. Tsang, On Roth's theorem concerning a cube and three cubes of primes, to appear in Quart. J. Math. (Oxford),55(2004),357-374.
[8]X. Ren and K. M. Tsang, On representation of integers by sums of a cube and three cubes of primes, Michigan Math. J., (3)53(2005),571-577.
[9]J. Y. Liu and T. Zhan, New development in the additive theory of prime Numbers, to be printed.
[10]L. K. Hua, Additive theory of prime numbers(in Chinese), Science Press, Beijing,1957; English transl., Amer. Math. Soc. Providence, RI,1965.
[11]R. C. Vaughan, The Hardy-Littlewood method,2nd Edition, Cambridge University Press, Cambridge,1997,11-12,104-105.
[12]H. Iwaniec, E. Kowalski, Analytic number theory, Amer. Math. Sco. Colloq. Publ. vol.53, Amer. Math. Providence,2004.
[13]Vinogradov, I. M., The method of trigonometric sums in number theory, in Selected works, Springer,1984.
[14]Gallagher, P. X., A large sieve density estimate near σ= 1, Invent. Math.11(1970), 329-339.