Dirichlet L-函数、Gauss和及其应用

发布时间：2018-07-08 10:15

本文选题：Dirichlet + L-函数　；参考：《西北大学》2016年博士论文

【摘要】：本文运用DirichletL-函数、Gauss和的相关性质研究了一些重要和式的均值估计问题,其中包括指数和、Ramanujan和、Dedekind和、Kloosterman和以及它们的各种推广和式.此外,还研究了特殊原根的存在性,特殊整数的表法问题等.具体说来,本文的主要成果如下：1.指数和的研究在近代解析数论的发展中起到了非常重要的作用.本文研究了二项指数和及三项指数和的均值问题,给出了三次二项指数和的四次均值和六次均值的精确公式,三项指数和的四次均值的精确公式；研究了二项指数和与Dedekind和、特征和多项式之间的关系,并得到了一些渐近公式.2.与Ramanujan和有关的混合均值问题研究.本文主要利用初等及解析的方法研究了Ramanujan和与Dedekind和、Hardy和的混合均值问题,给出了确切的计算公式.3.关于Kloosterman和的研究有着悠久的历史和丰富的内容.研究了经典Kloosterman和与2维Kloosterman和的混合均值问题；利用Gauss和的性质与广义指数和的估计,研究了高维类Kloosterman和的上界估计问题；并通过研究Gauss和的性质,Dirichlet L-函数的均值定理以及相关性质,得到了Kloosterman和与Dedekind和、Hardy和以及D. H. Lehmer问题的混合均值的计算公式.4. Dirichlet L-函数、Gauss和的应用.研究了一些特殊原根的存在性,给出了满足某些特定条件的原根个数的渐近公式；并研究了特征和的估计以及特殊整数的表法问题.
[Abstract]:In this paper, we use the properties of the Dirichlet L- function Gauss sum to study the mean value estimation of some important sums, including the exponential sum Ramanujan and the Dedekind sum Kloosterman sum and their generalized sums. In addition, the existence of special primitive roots and the representation of special integers are studied. Specifically, the main results of this paper are as follows: 1. The study of exponential sum plays a very important role in the development of modern analytic number theory. In this paper, we study the mean value of the binomial exponential sum and the cubic exponential sum, give the exact formulas of the fourth and sixth order mean of the cubic binomial exponential sum, the exact formula of the fourth power mean of the triple exponential sum, and study the binomial index sum and the Dedekind sum. Some asymptotic formulas. 2. Study on the mixed mean value problem with Ramanujan and related. In this paper, the mixed mean value problem of Ramanujan sum and Dedekind sum Hardy sum is studied by means of elementary and analytical methods, and the exact formula .3is given. The study of Kloosterman sum has a long history and rich contents. In this paper, the mixed mean value problem of classical Kloosterman sum and 2-dimensional Kloosterman sum is studied, and the upper bound estimation problem of Kloosterman sum is studied by using the properties of Gauss sum and the estimate of generalized exponential sum. By studying the properties of Gauss sum and the mean value theorem of Dirichlet L- function and its related properties, the formula of mixed mean value of Kloosterman sum and Dedekind sum Hardy sum and D. H. Lehmer problem is obtained. The application of Dirichlet L-function Gauss sum. In this paper, we study the existence of some special primitive roots, give the asymptotic formulas of the number of primitive roots satisfying some special conditions, and study the estimation of the characteristic sum and the problem of the representation of special integers.
【学位授予单位】：西北大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O156.4

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