A special case of Vinogradov's mean value theorem
A special case of Vinogradov's mean value theorem
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维诺格拉多夫中值定理的一个特例
DOI：
10.4064/aa-79-3-193-204
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发表时间：
1997
期刊：
影响因子：
0.7
通讯作者：
T. Wooley
中科院分区：
文献类型：
--
作者：
R. Vaughan;T. Wooley
i=1（xji-yi）= 0（1 ≤ j ≤ k），其中xi，yi ∈ [1，P ]<$Z具有很大的实用性。维诺格拉多夫在本世纪上半叶的开创性著作也许最能说明这一点（例如，见[1，6]）。尽管现代的发展，这样的估计仍然是主要的工具，在建立最知名的结果关于零区域的黎曼zeta函数，和最小数量G（k）的变量，其中渐近公式持有华林的问题。当s k + 1时，后一个渐近公式似乎远远超出了当前技术的掌握。我们在本书中的主要目的是以一种相当精确的形式建立s = k + 1情形下所需的渐近公式。当s是自然数时，令Ts（P）表示s元组x和y的数目，其中1 ≤ xi，yi ≤ P（1 ≤ i ≤ s），并且xi是yj的置换，因此特别地，Ts（P）= s！P s + Os（P s−1）。在第二节中，我们建立了渐近公式Jk+1，k（P）<$Tk+1（P）的强形式，并定义了（1.2）αn = min 1≤r≤ nr ∈N（r + n/r）.
i=1 (xji − y i ) = 0 (1 ≤ j ≤ k) with xi, yi ∈ [1, P ] ∩ Z are of great utility. This is perhaps best illustrated by the seminal works of Vinogradov from the first half of this century (see, for example, [1, 6]). Despite modern developments, such estimates remain the primary tool in establishing the best known results concerning the zerofree region of the Riemann zeta function, and the smallest number G(k) of variables for which the asymptotic formula holds in Waring’s problem. When s k + 1 the latter asymptotic formula seems far beyond the grasp of current technology. Our primary purpose in this memoir is to establish in a rather sharp form the desired asymptotic formula in the case s = k + 1. When s is a natural number, let Ts(P ) denote the number of s-tuples x and y in which 1 ≤ xi, yi ≤ P (1 ≤ i ≤ s), and the xi are a permutation of the yj , so that in particular, Ts(P ) = s!P s + Os(P s−1). In Section 2 we establish the strong form below of the asymptotic formula Jk+1,k(P ) ∼ Tk+1(P ), and in connection with this we define (1.2) αn = min 1≤r≤n r∈N (r + n/r).