本文選題：Nevanlinna理論　切入點：亞純函數(shù)　出處：《山東大學》2017年博士論文

【摘要】：作為千禧難題之一的黎曼猜想,長期以來備受許多數(shù)學工作者們的關注.1989年,Selberg為了研究L-函數(shù)的線性組合的值分布,以Riemann zeta函數(shù)為原型,定義了一類Dirichlet級數(shù),其滿足歐拉乘積,解析延拓,Riemann-型函數(shù)方程,且提出了關于這一類函數(shù)的幾個基本猜想.引人興趣的是,Selberg指出這些猜想緊密聯(lián)系著數(shù)論中的某些相關的經(jīng)典猜想.從此而后,這一類所謂的Selberg類L-函數(shù)成為了復分析理論中的另一個非常熱門的研究課題,也是現(xiàn)代解析數(shù)論中的重要研究對象,但是目前對于這一類函數(shù)的理解尚未達到一個完整的框架.事實上,Selberg猜測,黎曼假設對所有Selberg類中的函數(shù)L成立.由黎曼猜想衍生出來的一類重要問題是關于簡單零點在全部非平凡零點中所占比例的估計.數(shù)學家們曾普遍猜測,函數(shù)L的所有零點都是簡單零點,我們稱之為簡單零點假設.但此命題迄今尚未得到證明.不過,與黎曼猜想類似,簡單零點假設也得到了許多數(shù)值及解析結果的支持.Steuding在文[54]中給出了關于廣義Selberg類L-函數(shù)c值點的漸進公式,并將其應用到Nevanlinna值分布理論上.此方向引起了許多學者的興趣,對此進行了深入研究,成功地將兩個交叉學科融合在一起.最近,扈和李在文[35]中利用Riemann zeta函數(shù)在臨界直線上的零點構造了一個整函數(shù),并利用此函數(shù)將黎曼猜想轉(zhuǎn)換成亞純函數(shù)的唯一性問題.本文以Nevanlinna值分布理論為主要研究工具,討論了廣義Selberg類L-函數(shù)的零點分布問題和唯一性問題.文章共分如下五章:第一章為預備知識.簡要介紹了 Selberg類L-函數(shù)的基礎知識和Nevan-linna基本理論.第二章,研究了 Dirichlet L-函數(shù)的單零點分布問題.借助值分布理論,結合函數(shù)論中的abc猜想定理,給出了關于模k的一族Dirichlet L-函數(shù)的判別零點估計式.此外,證明了對任意有窮復數(shù)a,L-a的單零點在其全部零點中所占的比例是個正值,至多除掉兩個例外值,并且給出了此比例值的下確界.第三章,討論了廣義Selberg類L-函數(shù)的導函數(shù)L(k)(s))的零點分布問題.首先給出了L(k)(s)左右兩側的非零區(qū)域,并進一步給出L(k)(s)的零點估計式.第四章,探討了廣義Selberg類L-函數(shù)具有分擔集合的唯一性問題,推廣了 Steuding[55]和李[43]的結果.第五章,研究了廣義Selberg類L-函數(shù)與亞純函數(shù)具有分擔值的唯一性問題.文中結果推廣了李[42],Garunkstis,Grahl和Steuding[22]的結果.
[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].
【學位授予單位】：山東大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O174

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