关于L-函数的若干问题

发布时间：2018-03-27 05:35

本文选题：Nevanlinna理论　切入点：亚纯函数　出处：《山东大学》2017年博士论文

【摘要】：作为千禧难题之一的黎曼猜想,长期以来备受许多数学工作者们的关注.1989年,Selberg为了研究L-函数的线性组合的值分布,以Riemann zeta函数为原型,定义了一类Dirichlet级数,其满足欧拉乘积,解析延拓,Riemann-型函数方程,且提出了关于这一类函数的几个基本猜想.引人兴趣的是,Selberg指出这些猜想紧密联系着数论中的某些相关的经典猜想.从此而后,这一类所谓的Selberg类L-函数成为了复分析理论中的另一个非常热门的研究课题,也是现代解析数论中的重要研究对象,但是目前对于这一类函数的理解尚未达到一个完整的框架.事实上,Selberg猜测,黎曼假设对所有Selberg类中的函数L成立.由黎曼猜想衍生出来的一类重要问题是关于简单零点在全部非平凡零点中所占比例的估计.数学家们曾普遍猜测,函数L的所有零点都是简单零点,我们称之为简单零点假设.但此命题迄今尚未得到证明.不过,与黎曼猜想类似,简单零点假设也得到了许多数值及解析结果的支持.Steuding在文[54]中给出了关于广义Selberg类L-函数c值点的渐进公式,并将其应用到Nevanlinna值分布理论上.此方向引起了许多学者的兴趣,对此进行了深入研究,成功地将两个交叉学科融合在一起.最近,扈和李在文[35]中利用Riemann zeta函数在临界直线上的零点构造了一个整函数,并利用此函数将黎曼猜想转换成亚纯函数的唯一性问题.本文以Nevanlinna值分布理论为主要研究工具,讨论了广义Selberg类L-函数的零点分布问题和唯一性问题.文章共分如下五章:第一章为预备知识.简要介绍了 Selberg类L-函数的基础知识和Nevan-linna基本理论.第二章,研究了 Dirichlet L-函数的单零点分布问题.借助值分布理论,结合函数论中的abc猜想定理,给出了关于模k的一族Dirichlet L-函数的判别零点估计式.此外,证明了对任意有穷复数a,L-a的单零点在其全部零点中所占的比例是个正值,至多除掉两个例外值,并且给出了此比例值的下确界.第三章,讨论了广义Selberg类L-函数的导函数L(k)(s))的零点分布问题.首先给出了L(k)(s)左右两侧的非零区域,并进一步给出L(k)(s)的零点估计式.第四章,探讨了广义Selberg类L-函数具有分担集合的唯一性问题,推广了 Steuding[55]和李[43]的结果.第五章,研究了广义Selberg类L-函数与亚纯函数具有分担值的唯一性问题.文中结果推广了李[42],Garunkstis,Grahl和Steuding[22]的结果.
[Abstract]:Riemann conjecture, one of the millennials conundrum, has long been concerned by many mathematics workers. In 1989, in order to study the value distribution of linear combination of L- functions, Riemann zeta function was used as the prototype to define a class of Dirichlet series, which satisfies the Euler product. In this paper, some basic conjectures about this kind of functions are put forward. What is interesting is that Selberg points out that these conjectures are closely related to some classical conjectures in number theory. This kind of so-called Selberg class L- function has become another very hot research topic in the theory of complex analysis, and it is also an important research object in modern analytic number theory. But the understanding of this type of function has not yet reached a complete framework. In fact, Selberg conjectured, Riemannian hypothesis holds for functions L in all Selberg classes. An important problem derived from Riemannian conjecture is the estimation of the proportion of simple zeros in all nontrivial zero points. Mathematicians have generally conjectured, All zeros of the function L are simple zeros, which we call the simple zero hypothesis. However, this proposition has not been proved so far. However, similar to Riemann's conjecture, The simple zeros hypothesis is also supported by many numerical and analytical results. In [54], the asymptotic formula for the C-valued points of generalized Selberg class L-functions is given and applied to the theory of Nevanlinna value distribution. This direction has attracted the interest of many scholars. In this paper, we have carried out an in-depth study and successfully fused the two interdisciplinary disciplines. Recently, Hu and Li Zaiwen constructed an entire function by using the Riemann zeta function at the zero point on a critical line. By using this function, Riemann's conjecture is transformed into the uniqueness problem of meromorphic functions. In this paper, the Nevanlinna value distribution theory is used as the main research tool. In this paper, we discuss the problem of zero point distribution and uniqueness of generalized Selberg class L- functions. This paper is divided into five chapters as follows: chapter 1 is preparatory knowledge. The basic knowledge of Selberg class L- functions and the basic theory of Nevan-linna are briefly introduced. In this paper, the problem of single zero distribution of Dirichlet L- functions is studied. With the help of the theory of value distribution and the abc conjecture theorem in function theory, the estimators of a family of Dirichlet L- functions for module k are given. It is proved that the proportion of single zeros in all zeros of an arbitrary finite complex number aqila is positive, and at most two exceptional values are eliminated, and the lower bound of the ratio value is given in chapter 3. In this paper, we discuss the problem of the distribution of the zero point of the derivative function of the generalized Selberg class L-). First, we give the non-zero region on the left and right sides of the L ~ (+) K ~ ((+)), and further give the zero estimation formula of the L ~ (+) K ~ (+ +). In this paper, we discuss the uniqueness of generalized Selberg class L- functions with shared sets, and generalize the results of Steuding [55] and Li [43]. In this paper, we study the uniqueness of generalized Selberg class L- functions and meromorphic functions with shared values. In this paper, we generalize the results of Li [42] Garunkstis Grahl and Steuding [22].
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O174

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