On Weyl sums over primes and almost primes
On Weyl sums over primes and almost primes
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DOI：
10.1307/mmj/1156345592
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发表时间：
2006-08
影响因子：
0.9
通讯作者：
A. Kumchev
中科院分区：
文献类型：
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作者：
A. Kumchev
其中α是实数，k是正整数，e(Z)=exp(2πiz)，并且求和超过素数。这个和是由I.M.Vinogradov在20世纪30年代末的S作为解析数论中的一个工具引入的。1937年，Vinogradov发展了一种巧妙的新方法来估计素数上的和，并应用该方法得到了k=1的f(α)的第一个无条件估计。这个估计是他著名的证明[25]中的主要新奇之处，即每个足够大的奇数都是三个素数的和。在[27，第6章]中给出的更尖锐的形式中，Vinogradov的结果指出(本质上)如果a和q是满足
where α is a real number, k is a positive integer, e(z) = exp(2πiz), and the summation is over prime numbers. This sum was introduced as a tool in analytic number theory by I. M. Vinogradov in the late 1930’s. In 1937, Vinogradov developed an ingenious new method for estimating sums over primes and applied that method to obtain the first unconditional estimate for f(α) with k = 1. That estimate is the main novelty in his celebrated proof [25] that every sufficiently large odd integer is the sum of three primes. In the sharper form given in [27, Chapter 6], Vinogradov’s result states (essentially) that if a and q are integers satisfying