Estudamos a Torção Analítica para variedades com bordo e ainda com singuaridades do tipo cônico, mais especificamente, para um cone métrico limitado, com o propósito de investigar a extensão natural do Teorema de Cheeger Müller para tais espaços. Começamos determinando a Torção Analítica do disco e de variedades com o bordo totalmente geodésico, por meio de ferramentas geométricas desenvolvidas por J. Brüning e X. Ma. Posteriormente, usando ferramentas analíticas desenvolvidas por M. Spreafico, determinamos a Torção Analítica do cone sobre uma esfera de dimensão ímpar e provamos um teorema do tipo Cheeger Müller para este espaço. Mais ainda, provamos que o resualto de J. Brüning e X. Ma estende para o cone sobre uma esfera de dimensão ímpar / We study for Analytic Torsion of manifolds with boundary and also with conical singularities , more specifically, for a finite metric cone, with the purpose of investing the natural extension of the Cheeger Müller theorem for such spaces. we start by computing the Analytic Torsion of an any dimensional disc and of a manifold with totally boundary, by using geometric tools development by J. Brüning and X. Ma. Then, by using analytic tools development by M. Spreafico, we determine the Analytic Torsion of a cone over an odd dimensional sphere and we prove a theorem of Cheeger Müller type space. Moreover, we prove that the result of J. Brüning and X. Ma extends to the cone over an odd dimensional sphere

This thesis in analytic number theory consists of 3 parts and 13 individual papers. In the first part we prove some results in Turán power sum theory. We solve a problem of Paul Erdös and disprove conjectures of Paul Turán and K. Ramachandra that would have implied important results on the Riemann zeta function. In the second part we prove some new results on moments of the Hurwitz and Lerch zeta functions (generalized versions of the Riemann zeta function) on the critical line. In the third and final part we consider the following question: What is the natural generalization of the classical Poisson summation formula from the Fourier analysis of the real line to the matrix group SL(2,R)? There are candidates in the literature such as the pre-trace formula and the Selberg trace formula. We develop a new summation formula for sums over the matrix group SL(2,Z) which we propose as a candidate for the title "The Poisson summation formula for SL(2,Z)". The summation formula allows us to express a sum over SL(2,Z) of smooth functions f on SL(2,R) with compact support, in terms of spectral theory coming from the full modular group, such as Maass wave forms, holomorphic cusp forms and the Eisenstein series. In contrast, the pre-trace formula allows us to get such a result only if we assume that f is also SO(2) bi-invariant. We indicate the summation formula's relationship with additive divisor problems and the fourth power moment of the Riemann zeta function as given by Motohashi. We prove some identities on Kloosterman sums, and generalize our main summation formula to a summation formula over integer matrices of fixed determinant D. We then deduce some consequences, such as the Kuznetsov summation formula, the Eichler-Selberg trace formula and the classical Selberg trace formula.

In the thesis, the joint universality of periodic Hurwitz zeta-functions as well as that jointly with the Riemann zeta-functions of normalized cusp forms is obtained. / Darbe yra įrodomas jungtinis universalumas periodinėms Hurvico dzeta funkcijoms, taip pat bendras universalumas su Rymano dzeta funkcija ir normuotų parabolinių formų dzeta funkcija.

Darbe yra įrodomas jungtinis universalumas periodinėms Hurvico dzeta funkcijoms, taip pat bendras universalumas su Rymano dzeta funkcija ir normuotų parabolinių formų dzeta funkcija. / In the thesis, the joint universality of periodic Hurwitz zeta-functions as well as that jointly with the Riemann zeta-function or zeta functions of normalized cusp forms is obtained.

Darbe nagrinėjamos Hurvico dzeta funkcijos _dzeta(s; alfa_), s = _alfa +it su algebriniu iracionaliuoju parametru _alfa, 0 < alfa_ ≤ 1 diskretusis reikšmių pasiskirstymas. Įrodyta, jog funkcijai _(s; alfa_) galioja diskrečioji ribinė teorema su svoriu kompleksinėje plokštumoje C. / Master’s work is devoted to the investigation of value distribution of Hurwitz zeta-function _(s; alfa_), s = alfa_ + it with algebraic irrational parameter alfa_, 0 < alfa_ ≤ 1. It is proved that for the function _(s; alfa_) valid discrete limit theorem with weight in the complex plane.

Magistro darbe nagrinėjamos periodinės dzeta funkcijos ir periodinės Hurvico dzeta funkcijos jungtinis reikšmių pasiskirstymas ir jungtinė ribinė teorema tikimybinių matų silpno konvergavimo prasme kompleksinėje plokštumoje. / Master's thesis the periodic zeta functions and zeta functions of periodic Hurwitz joint distribution of the values ​​and the joint limit theorem of probability measures converge weak sense of the complex plane.

No description available.

We consider the regularization of some equivariant Euler classes of certain infinite-dimensional vector bundles over a finite-dimensional manifold M using the framework of zeta-regularized products [35, 53, 59]. An example of such a regularization is the Atiyah–Witten regularization of the T-equivariant Euler class of the normal bundle v(TM) of M in the free loop space LM [2]. In this thesis, we propose a new regularization procedure — W-regularization — which can be shown to reduce to the Atiyah–Witten regularization when applied to the case of v(TM). This new regularization yields a new multiplicative genus (in the sense of Hirzebruch [26]) — the ^Γ-genus — when applied to the more general case of a complex spin vector bundle of complex rank ≥ 2 over M, as opposed to the case of the complexification of TM for the Atiyah–Witten regularization. Some of its properties are investigated and some tantalizing connections to other areas of mathematics are also discussed. We also consider the application of W-regularization to the regularization of T²- equivariant Euler classes associated to the case of the double free loop space LLM. We find that the theory of zeta-regularized products, as set out by Jorgenson–Lang [35], Quine et al [53] and Voros [59], amongst others, provides a good framework for comparing the regularizations that have been considered so far. In particular, it reveals relations between some of the genera that appeared in elliptic cohomology, allowing us to clarify and prove an assertion of Liu [44] on the ˆΘ-genus, as well as to recover the Witten genus. The ^Γ₂-genus, a new genus generated by a function based on Barnes’ double gamma function [5, 6], is also derived in a similar way to the ^Γ-genus. / Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008

We consider the regularization of some equivariant Euler classes of certain infinite-dimensional vector bundles over a finite-dimensional manifold M using the framework of zeta-regularized products [35, 53, 59]. An example of such a regularization is the Atiyah–Witten regularization of the T-equivariant Euler class of the normal bundle v(TM) of M in the free loop space LM [2]. In this thesis, we propose a new regularization procedure — W-regularization — which can be shown to reduce to the Atiyah–Witten regularization when applied to the case of v(TM). This new regularization yields a new multiplicative genus (in the sense of Hirzebruch [26]) — the ^Γ-genus — when applied to the more general case of a complex spin vector bundle of complex rank ≥ 2 over M, as opposed to the case of the complexification of TM for the Atiyah–Witten regularization. Some of its properties are investigated and some tantalizing connections to other areas of mathematics are also discussed. We also consider the application of W-regularization to the regularization of T²- equivariant Euler classes associated to the case of the double free loop space LLM. We find that the theory of zeta-regularized products, as set out by Jorgenson–Lang [35], Quine et al [53] and Voros [59], amongst others, provides a good framework for comparing the regularizations that have been considered so far. In particular, it reveals relations between some of the genera that appeared in elliptic cohomology, allowing us to clarify and prove an assertion of Liu [44] on the ˆΘ-genus, as well as to recover the Witten genus. The ^Γ₂-genus, a new genus generated by a function based on Barnes’ double gamma function [5, 6], is also derived in a similar way to the ^Γ-genus. / Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008

The Hilbert-P\'olya conjecture asserts that the non-trivial zeros of the Riemann zeta function $\zeta(s)$ correspond (in a certain canonical way) to the eigenvalues of some positive operator. R. Meyer constructed a differential operator $D_-$ acting on a function space $\H$ and showed that the eigenvalues of the adjoint of $D_-$ are exactly the nontrivial zeros of $\zeta(s)$ with multiplicity correspondence. We follow Meyer's construction with a slight modification. Specifically, we define two function spaces $\H_\cap$ and $\H_-$ on $(0,\infty)$ and characterize them via the Mellin transform. This allows us to show that $Z\H_\cap\subseteq\H_-$ where $Zf(x)=\sum_{n=1}^\infty f(nx)$. Also, the differential operator $D$ given by $Df(x)=-xf'(x)$ induces an operator $D_-$ on the quotient space $\H=\H_-/Z\H_\cap$. We show that the eigenvalues of $D_-$ on $\H$ are exactly the nontrivial zeros of $\zeta(s)$. Moreover, the geometric multiplicity of each eigenvalue is one and the algebraic multiplicity of each eigenvalue is its vanishing order as a nontrivial zero of $\zeta(s)$. We generalize our construction on the Riemann zeta function to some $L$-functions, including the Dirichlet $L$-functions and $L$-functions associated with newforms in $\mathcal S_k(\Gamma_0(M))$ with $M\ge1$ and $k$ being a positive even integer. We give spectral interpretations for these $L$-functions in a similar fashion.