广义Dedekind和与二项指数和的混合均值及一类丢番图方程解的问题
发布时间：2019-06-21 02:00
【摘要】：数论是研究整数的各种性质的学科,具有“数学皇冠”称号的数论是数学领域中最优美的分支.作为高度抽象的学科,它的许多问题的研究和解决有着深远的意义.其中关于数论中一些著名和式的均值分布问题一直是数论研究的核心内容.本文主要是利用初等方法和解析方法对广义Dedekind和与二项指数和的混合均值及一类数论函数方程的解的问题进行了研究,具体研究成果包括以下几方面的内容：1.通过广义Dedekind和,二项指数和与Dirichlet L-函数问的关系,利用特征和估计及Dirichlet级数研究了广义Dedekind和与二项指数和的混合均值,并给出了一些较强的渐近公式.形如：2.通过对指数Diophantine方程的研究,利用初等方法和Pell方程的最小解的特性,给出了当x,y是奇素数时,方程xz的正整数解的情况.
[Abstract]:Number theory is a subject that studies the various properties of integers, and the number theory with the title of "mathematical crown" is the most beautiful branch in the field of mathematics. As a highly abstract subject, the research and solution of many of its problems is of far-reaching significance. Among them, the mean distribution of some famous sums in number theory has always been the core content of number theory. In this paper, the mixed mean value of generalized Dedekind sum and binomial exponential sum and the solution of a class of number theory function equations are studied by using elementary method and analytical method. The concrete research results include the following aspects: 1. By using the relation between generalized Dedekind sum, binomial exponential sum and Dirichlet L-function, the mixed mean value of generalized Dedekind sum and binomial exponential sum is studied by using characteristic sum estimation and Dirichlet series, and some strong asymptotic formulas are given. In the form of: 2. By studying the exponential Diophantine equation, using the elementary method and the characteristics of the minimum solution of the Pell equation, the case of the positive integer solution of the equation xz is given when x, y is an odd prime.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O156
本文编号：2503698
[Abstract]:Number theory is a subject that studies the various properties of integers, and the number theory with the title of "mathematical crown" is the most beautiful branch in the field of mathematics. As a highly abstract subject, the research and solution of many of its problems is of far-reaching significance. Among them, the mean distribution of some famous sums in number theory has always been the core content of number theory. In this paper, the mixed mean value of generalized Dedekind sum and binomial exponential sum and the solution of a class of number theory function equations are studied by using elementary method and analytical method. The concrete research results include the following aspects: 1. By using the relation between generalized Dedekind sum, binomial exponential sum and Dirichlet L-function, the mixed mean value of generalized Dedekind sum and binomial exponential sum is studied by using characteristic sum estimation and Dirichlet series, and some strong asymptotic formulas are given. In the form of: 2. By studying the exponential Diophantine equation, using the elementary method and the characteristics of the minimum solution of the Pell equation, the case of the positive integer solution of the equation xz is given when x, y is an odd prime.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O156
【参考文献】
相关期刊论文 前7条
1 ToddCochrane ,郑志勇;Upper bounds on a two-term exponential sum[J];Science in China,Ser.A;2001年08期
2 ;The mean value involving Dedekind sums and two-term exponential sums[J];Science China(Mathematics);2012年03期
3 张文鹏;ON THE FOURTH POWER MEAN OF THE DEDEKIND SUMS[J];宁夏大学学报(自然科学版);1999年01期
4 谢敏,张文鹏;关于广义Dedekind和的2k次均值[J];数学学报;2001年01期
5 张文鹏;广义Dedekind和与L-函数的一类恒等式[J];数学学报;2001年02期
6 刘燕妮;郭晓艳;;一个丢番图方程及其它的整数解[J];数学学报;2010年05期
7 张中峰;罗家贵;袁平之;;关于丢番图方程x~y+y~x=z~z[J];数学年刊A辑(中文版);2013年03期
，本文编号：2503698
本文链接：https://www.wllwen.com/kejilunwen/yysx/2503698.html
