摘要
小区间内的华林-歌德巴赫问题吸引了许多作者并且被许多作者研究过，其中以哥德巴赫-维诺格拉多夫关于几乎相等的素数的定理最为著名(读者可见[1]，[2]，[3]和[4])。
不同于线性情形，研究非线性情形需克服更大的困难。具体说来，在应用圆法时需处理更大的主区间，
刘建亚和展涛[7]首先在广义黎曼猜想下证明了；每个模24余5的大整数N可以表成5个几乎相等的素数的平方之和，即
N=p_1~2+p_2~2+p_3~2+p_4~2+P_5~2，这里|p_j-(N/5)~(1/2)|≤U，j=1，2，…，5，(0.5)其中U=N~((1/2)-δ+ε)，δ=1/20。
1998年，刘和展在研究华林-哥德巴赫的问题中找到了处理扩大了的主区间的新方法。这一方法被成功地应用到许多关于素数的加性问题，例如[5]和[6]，当这一方法被应用到问题(0.5)时，如下的无条件结果被陆续得到：刘和展的δ=1/50，1/48([7]，[8])；Bauer的δ=19/180([9])；吕广世的δ=1/35([10])．
在这篇文章里我们将研究两个小区间内的华林-哥德巴赫问题。
首先，在第一章中，我们将研究华罗庚先生的关于一个素数和三个素数的平方和的定理在小区间内的情形(见[11])．定理1．1 对于每个充分大的模3不同于0的整数N，方程对于U=N~((1/2)-(1/25)+ε)有解。
我们的定理1．1基于下面的结论。
Waring-Goldbach problems in short intervals have appealed to many authors and have been investigated, among which the Goldbach-Vinogradov theorem with almost equal prime variables may be the most famous one (see for example [1], [2], [3] and
[4]).
Different from this 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 [15] first studied the quadratic case assuming the Generalized Riemann Hypothesis (GRH). More precisely, they showed that under GRH each large integer TV ≡ 5(mod 24) can be written as
N=p_1~2+p_2~2 + p_3~2+p_4~2+p_5~2 (0.1)
where
Later Bauer [16] unconditionally showed that the formula (0.1) holds true for U =N~((1/2)-δ), where δ ≥0 and its exact value depends on the constants in the Deuring-Heilbronn phenomenon, and is not numerically determined.
In 1998 Zhan and Liu [15] 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 Dcuring-Heilbronn phenomenon can be avoided. Due to this approach, they obtained that (0.1) is true for U = N~((1/2)-(1/(50)+ε). With the development of this approach, the exponent (1/2) —(1/50) has subsequently reduced to (1/2) — (19/850) by Bauer [9] and then to (1/2) - (1/35) by the first author of [17].
In this paper we study two Waring-Goldbach problems in short intervals.
引文
[1] A. Balog and A. Perelli, Exponential sums over primes in short intervals, Acta Math Hung, 48(1986), 223-228.
[2] C.D. Pan and C.B. Pan, On estimation of trigonometric sums over primes in short intervals(I), Sci. China, Ser.A, 32(1989), 408-416.
[3] T. Zhan, On the representation of large odd integer as a sum of three almost equal primes, Acta Mathematica Sinica, 7(3)1991, 259-272.
[4] Perelli A. and Pintz. J, On the exceptional set for Goldbach's problem in short intervals, J. London Math Soc, 47(1993), 41-49
[5] J.Y. Liu, and M.C. Liu, The exceptional set in four prime squares problem, Illinois J. Math., 44(2000), 272-293.
[6] C. Bauer, M.C. Liu and T. Zhan, On sums of three prime squares, J. Number Theory, 85(2000), 336-359.
[7] J.Y. Liu and T. Zhan, Sums of five almost equal prime squares, Sci. in China, Series A, 41(1998), 710-722.
[8] J.Y. Liu and T. Zhan, Hus's theorem on prime squares in short intervals, Acta Math. Sin., 16(2000), 1-22.
[9] C. Bauer, Sums of five almost equal prime squares, Acta Math Sin, 2005, 21: 883-840.
[10] G.S. Lu, Hua's Theorem with five almost equal prime variables, Chin. Ann. Math. Scr. B, (2005), 291-304.
[11] L.K. Hua, Some results in the additive prime number theory, Quart. J. Math. (Oxford), 9(1938), 68-80.
[12] J.Y. Liu and T. Zhan, Distributions of integers that are sums of three squares of primes, Acta Arith. 98(2001), no.3, 207-228.
[13] J.Y. Liu and T. Zhan, On sums of five almost equal prime squares, Acta Arith, 77(1996), 369-383.
[14] C.D. Pan and C.B. Pan, On estimation of trigonometric sums over primes in short intervals(Ⅲ), Chinese Ann. Math.(Ser. B), 11(1990), 138-147.
[15] J.Y. Liu and T. Zhan, On sums of five almost equal prime squares, Acta Arith, 77(1996), 369-383.
[16] C. Bauer, A note on sums of five almost equal prime squares, Arch.,Math., 69(1997), 20-30.
[17] G. S. Lu, Hua's Theorem with five almost equal prime variables, Chinese Annals of Mathematics, ser B, 2005, 26: 291-304.
[18] 孟宪萌，九个几乎相等的素数立方之和，山东大学学报，2002，37：31-37．
[19] 吕广世，九个几乎相等的素数立方之和，数学学报中文版，vol 49，No．1：195-204．
[20] J.Y. Liu, On Lagrange's theorem with prime variables, Quart. J. Math. (Oxford), 54(2003), 454-462.
[21] X. M. Ren, Exponential sums over primes,Sci China, Ser A, 2005, 35: 252-264.
[22] P.X. Gallagher, A large sieve density estimate near σ= 1, Invent. Math. 11(1970), 329-339.
[23] E.C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., University Press, Oxford 1986.
[24] C.D. Pan and C.B. Pan, Fundamentals of analytic number theory (in Chinese), Science Press, Beijing 1991.
[25] K. Prachar, Primzahlverteilung, Springer, Berlin 1957.
[26] H. Davenport, Multiplicative Number Theory, 2nd ed., Springer, Bcrlin 1980.
[27] M.N. Huxley, Large values of Dirichlet polynomials (Ⅲ), Acta Arith. 26(1974/75), 435-444.
[1] A. Balog and A. Perelli, Exponential sums over primes in short intervals, Acta Math Hung, 48(1986), 223-228.
[2] C.D. Pan and C.B. Pan, On estimation of trigonometric sums over primes in short intervals(I), Sci. China, Ser.A, 32(1989), 408-416.
[3] T. Zhan, On the representation of large odd integer as a sum of three almost equal primes, Acta Mathematica Sinica, 7(3)1991, 259-272.
[4] Perelli A. and Pintz. J, On the exceptional set for Goldbach's problem in short intervals, J. London Math Soc, 47(1993), 41-49.
[5] J.Y. Liu and T. Zhan, On sums of five almost equal prime squares, Acta Arith, 77(1996), 369-383.
[6] J.Y. Liu and T. Zhan, Sums of five almost equal prime squares, Sci. in China, Series A, 41(1998), 710-722.
[7] J.Y. Liu, and M.C. Liu, The exceptional set in four prime squares problem, Illinois J. Math., 44(2000), 272-293.
[8] C. Bauer, M.C. Liu and T. Zhan, On sums of three prime squares, J. Number Theory, 85(2000), 336-359.
[9] J.Y. Liu and T. Zhan, Hus's theorem on prime squares in short intervals, Acta .Math. Sin., 16(2000), 1-22.
[10] C. Bauer, Sums of five almost equal prime squares, Acta Math Sin, 2005, 21: 883-840.
[11] G.S. Lu, Hua's Theorem with five almost equal prime variables , Chin. Ann. Math. Ser. B, (2005), 291-304.
[12] L.K. Hua, Some results in the additive prime number theory, Quart. J. Math. (Oxford), 9(1938), 68-80.
[13] J.Y. Liu and T. Zhan, Distributions of integers that arc sums of three squares of primes, Acta Arith. 98(2001), no.3, 207-228.
[14] C.D. Pan and C.B. Pan, On estimation of trigonometric sums over primes in short intcrvals(Ⅲ), Chinese Ann. Math.(Scr. B), 11(1990), 138-147.
[15] J.Y. Liu and T. Zhan, On sums of five ahnost equal prime squares, Acta Arith, 77(1996), 369-383.
[16] C. Bauer, A note on sums of five almost equal prime squares, Arch. Math., 69(1997), 20-30.
[17] G. S. Lu, Hua's Theorem with five almost equal prime variables, Chinese Annals of Mathematics, set B, 2005, 26: 291-304.
[18] X. M. Meng, On sums of nine almost equal prime cubes, J. of Shandong University, 37(2002) 31-37.
[19] J.Y. Liu, On Lagrange's theorem with prime variables, Quart. J. Math. (Oxford), 54(2003), 454-462.
[20] X. M. Ren, Exponcntial sums over primes, Sci China, Set A, 2005, 35: 252-264.
[21] P.X. Gallagher, A large sieve density estimate near σ= 1, Invent. Math. 11(1970), 329-339.
[22] E.C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., University Press, Oxford 1986.
[23] C.D. Pan and C.B. Pan, Fundamentals of analytic number theory (in Chinese), Science Press, Beijing 1991.
[24] K. Prachar, Primzahlverteilung, Springer, Berlin 1957.
[25] H. Davenport, Multiplicative Number Theory, 2nd ed., Springer, Berlin 1980.
[26] M.N. Huxley, Large values of Dirichlct polynomials (Ⅲ), Acta Arith. 26(1974/75), 435-444.
[27] 吕广世．九个几乎相等的素数立方之和，数学学报中文版，vol 49，No．1：195-204．