在算术级数中具有算术函数系数的素变量指数和

发布时间：2018-05-18 18:17

本文选题：指数和 + 算术函数　；参考：《山东大学》2017年硕士论文

【摘要】：指数和是数论研究的核心课题,有重要的理论意义和应用价值.设集合M代表所有函数值为复数的积性函数的集合,M1(?)M且对(?)f ∈ M1,有性质|f(n)| ≤ 1,n ∈ 令e(α)= e2πiα,其中α代表无理数.令M'是算术函数的集合,其中函数符合一定的条件,例如M'=M1对于f ∈ M'或f为某一特殊的算术函数,本文主要研究形如指数和的渐进行为,其中E是一个整数集合并且1954年,Vinogradov[25]的一个非常著名的结果是如果Q(n)= k +αk-1nk-1+…+α1n是一个多项式,αkk,…,α1都是实数且最少有一个是无理数.那么1974年,Daboussi和Delange[6]证明了对于任意给定的无理数α,f∈ 1 式成立后来,Delange扩大了此结果的函数类,对于f ∈ L2,也就是对任意的f ∈M,满足条件结果是成立的.Indlekofer又将f扩展到更大的函数类里去,对f ∈ L*,即对任意的f ∈.M,满足0,结果仍然成立.1986 年,I.Katai[9]证明了2012年,J.M.De Koninck和I.Katai[17]定义了一种新的算术函数l(n):=g1(F1(n)… gs(Fs(n)),其中F1(n),…,Fs(n)是整系数多项式,并且只有当x0的时候,Fi(x)0,i = 1,2,…,s.gi(i = 1,2,…,s)都是复值可乘函数,并且满足特殊条件.令得到了两个新的结果:本文的主要工作是将上述素变量指数和推广到算术级数中素变量指数和中去,即证明定理0.1 对于固定的k,l 满足(k,l)=1,当x→∞时,有下式成立其中l(n):=g1(F1(n))….gs(Fs(n)),F1(x),…,s(x)∈ Z[x],并且只有当x0时,F1(x),…,Fs(x)0.对于i = 1,2,...,s,gi都是复值的可乘函数,并且满足特殊条件(在第三章中介绍).本文主要使用Turan—Kubilius不等式[24]的经典方法来证明我们的结论.定理的证明用到了 G.Tenenbaum[24]第三章的内容和J.M.De Koninck与I.Katai[17]的相关引理以及初等数论和解析数论的知识.本文共分三个章节,第一章是导言部分,主要介绍问题的研究背景和目前的研究成果;第二章主要做一些预备工作;第三章是论文的主体,讲的是论文证明过程中需要的引理以及本文主要定理的证明.
[Abstract]:The index and the core subject of the study of number theory has important theoretical significance and applied value. Set M represents the set of integrable functions of all functions as complex numbers, M1 (?) M and (?) f f M1, which has a property |f (n) or less than 1, n e (alpha) = E2 PI I alpha, which represents the irrational number. For example, M'=M1 for a special arithmetic function for F M'or F, this article mainly deals with exponential and progressive behavior, in which E is a set of integers and in 1954, a very famous result of Vinogradov[25] is if Q (n) = K + alpha k-1nk-1+... + alpha 1n is a polynomial, alpha KK,... And, alpha 1 is real and at least one is irrational. Then, in 1974, Daboussi and Delange[6] proved that after the establishment of any given irrational number alpha, F 1, Delange expanded the function class of the result, that is, f L2, that is, any f M, and the conditional result is that the.Indlekofer also extends f to a larger function. In the number of classes, the f L*, that is, to any f.M, satisfies 0, the result is still set up for.1986 years. I.Katai[9] proves that in 2012, J.M.De Koninck and I.Katai[17] defined a new arithmetic function L (n): =g1. GS (Fs (n)), where F1 (n),... Fs (n) is an integer coefficient polynomial, and only when x0, Fi (x) 0, I = 1,2,... S.gi (I = 1,2,... S) is a complex valued multiplicative function and satisfies special conditions. Two new results are obtained: the main work of this paper is to extend the above prime variable exponent and extend to the prime variable exponent and in the arithmetic progression. That is, theorem 0.1 is proved to be a fixed k, l satisfies (k, L) =1, and when X - infinity, there is a l (n): =g1 (F1 (n)). .gs (Fs (n)), F1 (x),... S (x) Z[x], and only when x0, F1 (x),... Fs (x) 0. is a multiplicative function of complex values for I = 1,2,..., s and GI, and satisfies special conditions (introduced in the third chapter). This paper mainly uses the classical method of Turan Kubilius inequality [24] to prove our conclusion. The proof of the theorem uses the contents of the G.Tenenbaum[24] third chapter and the related lemma of J.M.De Koninck and the J.M.De Koninck. As well as the knowledge of elementary number theory and analytic number theory, this article is divided into three chapters. The first chapter is the introduction, which mainly introduces the research background and the present research results; the second chapter mainly makes some preparatory work; the third chapter is the main body of the paper, which is about the lemma needed in the process of the paper and the proof of the main theorems in this paper.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O156

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