Slim Exceptional Sets For Sums of Four Squares
Slim Exceptional Sets For Sums of Four Squares
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DOI：
10.1112/plms/85.1.1
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发表时间：
2002-07
影响因子：
1.8
通讯作者：
T. Wooley
中科院分区：
文献类型：
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作者：
T. Wooley
鉴于现有技术允许人们建立几乎所有满足适当同余条件的自然数都被表示为质数的三个平方和，人们期望在涉及四个质数平方和的类似问题中的例外集可以获得强有力的估计。设E(N)表示不超过N且与4模24相等，但不能写成质数的四个平方和的正整数的个数。本文描述了一种方法，该方法表明，对于每一个正数，有E(N)≪N13/30+ λ，从而有效地利用了质数的“多余”四分之一平方，从而改善了J. Liu和M.‐C所提出的E(N)≪N13/15+ λ。刘。这表明，只要有足够多的过剩变量可用，这一进展背后的思想允许在各种可加性问题中对异常集的估计显着缩小。这些思想在涉及四个平方和的几个附加问题中得到了说明。
Given that available technology permits one to establish that almost all natural numbers satisfying appropriate congruence conditions are represented as the sum of three squares of prime numbers, one expects strong estimates to be attainable for exceptional sets in the analogous problem involving sums of four squares of primes. Let E(N) denote the number of positive integers not exceeding N that are congruent to 4 modulo 24, yet cannot be written as the sum of four squares of prime numbers. A method is described that shows that for each positive number ɛ, one has E(N)≪N13/30+ϵ , thereby exploiting effectively the ‘excess’ fourth square of a prime so as to improve the recent bound E(N)≪N13/15+ϵ due to J. Liu and M.‐C. Liu. It transpires that the ideas underlying this progress permit estimates for exceptional sets in a variety of additive problems to be significantly slimmed whenever sufficiently many excess variables are available. Such ideas are illustrated for several additional problems involving sums of four squares.