The spectral zeta function, introduced by Minakshisundaram and Pleijel in [36] and denoted by ζg(s), encodes important spectral information for the Laplacian on Riemannian manifolds. For instance, the important notions of the determinant of the Laplacian and Casimir energy are defined via the spectral zeta function. On homogeneous manifolds, it is known that the spectral zeta function is critical with respect to conformal metric perturbations, (see e.g Richardson ([47]) and Okikiolu ([41])). In this thesis, we compute a second variation formula of ζg(s) on closed homogeneous Riemannian manifolds under conformal metric perturbations. It is well known that the quadratic form corresponding to this second variation is given by a certain pseudodifferential operator that depends meromorphically on s. The symbol of this operator was analysed by Okikiolu in ([42]). We analyse it in more detail on homogeneous spaces, in particular on the spheres Sn. The case n = 3 is treated in great detail. In order to describe the second variation we introduce a certain distributional integral kernel, analyse its meromorphic properties and the pole structure. The Casimir energy defined as the finite part of ζg(-½) on the n-sphere and other points of ζg(s) are used to illustrate our results. The techniques employed are heat kernel asymptotics on Riemannian manifolds, the associated meromorphic continuation of the zeta function, harmonic analysis on spheres, and asymptotic analysis.

在這篇論文裡，我們主要在利用值分佈理論來探討兩個半純函數共三個值的基本性質以及在不同條件下的唯一性。 / In this thesis, we study the properties of two non-constant meromorphic functions sharing three values, and the uniqueness theorems under various conditions.

Thesis (Ph. D.)--Ohio State University, 2003. / Title from first page of PDF file. Document formatted into pages; contains x, 227 p.; also includes graphics Includes bibliographical references (p. 224-227). Available online via OhioLINK's ETD Center

A Julia set for a complex function 𝑓 is the set of all points in the complex plane where the iterates of 𝑓 do not form a normal family. A picture of the Julia set for a function can be generated with a computer by coloring pixels (which we consider to be small squares) based on the behavior of the point at the center of each pixel. We consider the accuracy of computer generated pictures of Julia sets. Such a picture is said to be accurate if each colored pixel actually contains some point in the Julia set. We extend previous work to show that the pictures generated by an algorithm for the family λe² are accurate, for appropriate choices of parameters in the algorithm. We observe that the Julia set for meromorphic functions with polynomial Schwarzian derivative is the closure of those points which go to infinity under iteration, and use this as a basis for an algorithm to generate pictures for such functions. A pixel in our algorithm will be colored if the center point becomes larger than some specified bound upon iteration. We show that using our algorithm, the pictures of Julia sets generated for the family λtan(z) for positive real λ are also accurate. We conclude with a cautionary example of a Julia set whose picture will be inaccurate for some apparently reasonable choices of parameters, demonstrating that some care must be exercised in using such algorithms. In general, more information about the nature of the function may be needed. / Ph. D.

在這篇論文裡，我們探討 $\mathcal{A}$ 類半純函數的值分佈基本理論。我們證明了每一個 $\mathcal{A}$ 類半純函數最多有兩個重值，而這個結果是最佳的情形。進而，我們證明若一個 $\mathcal{A}$ 類半純函數 $f$ 與其導數 $f^{(k)}$ 共非零的複數值，則 $f\equiv f^{(k)}$。 / In this thesis, we study the basic theory of value distribution of meromorphic function of class $\mathcal{A}$. We prove that every meromorphic function of class $\mathcal{A}$ has at most two multiple values and the result is sharp. Also, we prove that if a meromorphic function $f$ of class $\mathcal{A}$ and its derivative $f^{(k)}$ share a non-zero complex value, then $f\equiv f^{(k)}$.

Haymen猜測:對任意的超越半純函數 f(z)，f'(z)f(z)^n 取所有值無窮多次，其中至多只有一個例外值。這個著名的猜測，大部分的情形已被證明是正確的。另外，Hayman 證明 f'(z)-af(z)^n 取所有有限值無窮多次 ，其中 a 為一複數且 n≧5 的正整數。在本篇論文裡，我們將探討以小函數為係數的半純函數微分多項式之值分佈問題。並將Hayman的結果推廣至 f^{k}(z)f(z)^n 與 f^{k}(z)-af(z)^n 的情形。同時，我們也證明一些 A類半純函數與其導數的值分佈結果。 / A famous conjecture of Hayman says that if f(z) is a transcendental meromorphic function, then f'(z)f(z)^n assumes all finite values except possibly zero infinitely often. The conjecture was solved in most cases. Another result of Hayman says that f'(z)-af(z)^n, where n≧5 and a is a complex number, assumes all finite values infinitely often. In this thesis, we will study the value distribution of some differential polynomial in a meromorphic function with small functions as coefficents. In fact, we will generalize Hayman's results to the cases f^(k)(z)f(z)^n and f^(k)(z)-af(z)^n. Also, the value distribution of meromorphic functions of class A with their derivatives are obtained.

在這篇論文裡，我們利用值分佈理論來探討兩個半純函數何時會相等，進而了解兩個半純函數滿足某一函數方程時，其關係為何。最後，我們探討兩個半純函數共四個少函數且擁有少量的極點之相關性質。 / In this thesis, we use the theory of value distribution to study the uniqueness problems of two meromorphic functions and the relation of two meromorphic functions satisfying some functional equations. Also we prove some uniqueness results on two meromorphic functions with few poles sharing four small functions.

Cette thèse porte sur des propriétés arithmétiques des fonctions méromorphes et transcendantes d'une variable. Dans le chapitre 3, nous définissons des mesures de transcendance pour les fonctions holomorphes et méromorphes sur un domaine régulier de C puis nous majorons ces mesures en fonction de la distribution des petites valeurs de la fonction étudiée.Grâce aux théories de Nevanlinna et d'Ahlfors, nous étudions dans le chapitre 4 la distribution des petites valeurs de certaines classes de fonctions méromorphes sur D ou C afin d'obtenir pour celles-ci des majorations explicites de leurs mesures de transcendance. L'application principale de ce travail est l'obtention de nouveaux lemmes de zéros polynomiaux pour de grandes familles de fonctions méromorphes et en particulier pour les fonctions de Weierstrass et les fonctions fuchsiennes. Dans le chapitre 5, nous montrons que ces lemmes de zéros polynomiaux conduisent à des bornes logarithmiques du nombre de points algébriques de degré et hauteur bornée contenus dans les graphes des fonctions étudiées. / This PhD thesis is about some arithmetic properties of meromorphic functions of one variable.In chapter 3, we define the transcendental measures for holomorphic and meromorphic functions on a regular domain of C, then we obtain upper bounds of these measures in terms of the distribution of small values of the function.Thanks to the Nevanlinna and Ahlfors theories, we study in chapter 4 the distribution of small values of some classes of meromorphic functions on D or C in order to obtain explicit upper bounds of transcendental measures.The main application of this work is the demonstration of new polynomial zero estimates for large classes of meromorphic functions, in particular for Weierstrass functions and fuchsian functions.In chapter 5, we prove that polynomial zero estimates lead to logarithmic bounds of the number of algebraic points of bounded degree and height contained in the graph of the function.

We study pseudodifferential operators on a cylinder IR x B with cross section B that conical singularities. Configurations of that kind are the local model of cornere singularities with base spaces B. Operators A in our calculus are assumed to have symbols α which are meromorphic in the complex covariable with values in the space of all cone operators on B. In case α is dependent of the axial variable t ∈ IR, we show an explicit formula for solutions of the homogeneous equation. Each non-bjectivity point of the symbol in the complex plane corresponds to a finite-dimensional space of solutions. Moreover, we give a relative index formula.

在這篇論文裡，我們利用值分佈的理論來探討半純函數的共值與唯一性的問題，本文包含了以下的結果：將Jank與Terglane有關三個A類中的半純函數唯一性的結果推廣到任意q個半純函數的情形；證明了C. C. Yang的一個猜測；建構了一類半純函數恰有兩個虧值，而且算出它們的虧格；將 Nevanlinna 五個值的定理推廣至兩個半純函數部分共值的情形；探討純函數 與其導數的共值問題；最後，證明了兩個半純函數共四個值且重數皆不同的定 理。 / In this thesis, we study the sharing value problems and the uniqueness problems of meromorphic functions in the theory of value distribution. In fact, this thesis contains the following results: We generalize a unicity condition of three meromorphic functions given by Jank and Terglane in class A to the case of arbitrary q meromorphic functoins. An elementary proof of a conjecture of C. C. Yang is provided. We construct a class of meromorphic functions with exact two deficient values and their deficiencies are explicitly computed. We generalize the Nevanlinna's five-value theorem to the cases that two meromorphic functions partially share either five or more values, or five or more small functions. In each case, we formulate a way to measure how far these two meromorphic functions are from sharing either values or small functions, and use this measurement to prove a uniqueness theorem. Also, we prove some uniqueness theorems on entire functions that share a pair of values (a,-a) with their derivatives, which are reformulations of some important results about uniqueness of entire functions that share values with their derivatives. Finally, we prove that if two distinct non-constant meromorphic functions $f$ and $g$ share four distinct values a_1, a_2, a_3, a_4 DM such that each a_i-point is either a (p,q)-fold or (q,p)-fold point of f and g, then (p,q) is either (1,2) or (1,3) and f, g are in some particular forms.