Estimates for complete multiple exponential sums
Estimates for complete multiple exponential sums
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完整多重指数和的估计
DOI：
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发表时间：
2000
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通讯作者：
J. Loxton
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文献类型：
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作者：
J. Loxton
其中求和是对x模q的一组完整的残差进行的，并且eq（t）= e2πit/q。对这些和的研究很容易被在解析数论和其他地方的应用所激发。第一个重要的估计总和在一个变量出现在工作外尔（1916年）的均匀分布。这导致了货车德尔科普特的方法与应用的zeta函数，除数问题和其他问题的乘法数论。多个指数和首次出现在工作的爱泼斯坦zeta函数Titchmarsh（1934年）。（Graham和Kolesnik（1991）讨论了历史和最近的结果。另一方面，更直接的相关性，以下，哈代和利特尔伍德（1919年）发现了一种新的方法来解决问题的加法数论，如问题的华林和哥德巴赫。用这种方法处理主弧涉及完全指数和。(See（参见Vaughan（1981））。Deligne（1974）证明了Weil定理，结果表明，对于素数p，|S（f ; p）|≤（d− 1）npn/2，
where the sum is taken over a complete set of residues for x modulo q and eq(t) = e2πit/q. The study of these sums is readily motivated by applications in analytic number theory and elsewhere. The first important estimates for sums in one variable appear in the work of Weyl (1916) on uniform distribution. This led to van der Corput’s method with applications to the zeta function, the divisor problem and other problems in multiplicative number theory. Multiple exponential sums first appeared in work on the Epstein zeta function by Titchmarsh (1934). (Graham and Kolesnik (1991) discuss the history and recent results.) On the other hand, and of more immediate relevance to what follows, Hardy and Littlewood (1919) found a new method for tackling problems in additive number theory such as the problems of Waring and Goldbach. The treatment of the major arcs by this method involves complete exponential sums. (See, for example, Vaughan (1981).) As a consequence of his proof of the Weil conjectures, Deligne (1974) showed that, for a prime p, |S(f ; p)| ≤ (d− 1)npn/2,