【摘要】：數(shù)論是研究整數(shù)的各種性質(zhì)的學(xué)科,具有“數(shù)學(xué)皇冠”稱(chēng)號(hào)的數(shù)論是數(shù)學(xué)領(lǐng)域中最優(yōu)美的分支.作為高度抽象的學(xué)科,它的許多問(wèn)題的研究和解決有著深遠(yuǎn)的意義.其中關(guān)于數(shù)論中一些著名和式的均值分布問(wèn)題一直是數(shù)論研究的核心內(nèi)容.本文主要是利用初等方法和解析方法對(duì)廣義Dedekind和與二項(xiàng)指數(shù)和的混合均值及一類(lèi)數(shù)論函數(shù)方程的解的問(wèn)題進(jìn)行了研究,具體研究成果包括以下幾方面的內(nèi)容：1.通過(guò)廣義Dedekind和,二項(xiàng)指數(shù)和與Dirichlet L-函數(shù)問(wèn)的關(guān)系,利用特征和估計(jì)及Dirichlet級(jí)數(shù)研究了廣義Dedekind和與二項(xiàng)指數(shù)和的混合均值,并給出了一些較強(qiáng)的漸近公式.形如：2.通過(guò)對(duì)指數(shù)Diophantine方程的研究,利用初等方法和Pell方程的最小解的特性,給出了當(dāng)x,y是奇素?cái)?shù)時(shí),方程xz的正整數(shù)解的情況.
[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.
【學(xué)位授予單位】：西北大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：O156

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