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This thesis is concerned with finding meromorphic extensions to a half-plane containing zero for certain generating functions. In particular, we generalise a result due to Morita and use it to show that the zeta function associated to the geodesic flow over a quotient of a Schottky group can be meromorphically extended to a half-plane containing zero. Moreover, we show that the special value at zero can be calculated. These results are then generalised to obtain meromorphic extensions past zero for L-functions defined on quotients of Schottky groups and to provide an expression for the special value at zero. Finally we show that Morita's method can be adapted to provide a meromorphic extension to a half-plane containing zero for Poincaré series defined for a Schottky group, and that in special circumstances the value at zero can be calculated.

The aim of this thesis is to examine the connection between the Riemannhypothesis and the distribution of prime numbers. We first derive theanalytic continuation of the zeta function and prove some of its propertiesusing a functional equation. Results from complex analysis such asJensen’s formula and Hadamard factorization are introduced to facilitatea deeper investigation of the zeros of the zeta function. Subsequently, therelation between these zeros and the asymptotic distribution of primesis rendered explicit: they determine the error term when the prime-counting function π(x) is approximated by the logarithmic integral li(x).We show that this absolute error is O(x exp(−c√log x) ) and that the Riemannhypothesis implies the significantly improved upper bound O(√x log x).

Disertaciją sudaro mokslinių tyrimų medžiaga, kurie atlikti 2008 -2012 metais Vilniaus universitete Matematikos ir informatikos fakultete. Disertacijoje įrodomos naujos teoremos apie Lercho ir Selbergo dzeta funkcijų reikšmių pasiskirstymą, atliekami kompiuteriniai skaičiavimai matematine programa MATHEMATICA. Disertaciją sudaro įvadas, 3 skyriai, išvados ir literatūros sąrašas. Disertacijos rezultatai atspausdinti trijuose moksliniuose straipsniuose, Lietuvos ir užsienio žurnaluose, pristatyti Lietuvoje ir užsienyje vykusiose mokslinėse konferencijose bei katedros seminarų metu. Pirmajame skyriuje įrodinėjamos ribinės teoremos Lercho dzeta funkcijai. Praėjusio šimtmečio ketvirtame dešimtmetyje Selbergas įrodė, kad tinkamai normuotas Rymano dzeta funkcijos logaritmas ant kritinės tiesės turi standartinį normalųjį pasiskirstymą. Selbergo įrodymas rėmėsi Oilerio sandauga, kuria turi Rymano dzeta funkcija, bet bendru atveju jos neturi Lercho dzeta funkcija. Antrajame skyriuje įrodoma teorema apie Lercho transcendentinės funkcijos nulių įvertį vertikaliose kompleksinės plokštumos juostose bei atliekami kompiuteriniai nulių skaičiavimai srityje Re(s)>1 programa MATHEMATICA. Trečiajame skyriuje nagrinėjamos dviejų Selbergo dzeta funkcijų monotoniškumo savybės, kurios yra tiesiogiai susijusios su šių funkcijų nulių išsidėstymu kritinėje juostoje. Monotoniškumo savybės lyginamos su Rymano dzeta funkcijos monotoniškumo savybėmis ir nulių išsidėstymu, kuris yra viena didžiausių... [toliau žr. visą tekstą] / The doctoral dissertation contains the material of scientific investigations done in 2008-2012 in the Faculty of Mathematics and Informatics at Vilnius University. The dissertation includes new theorems for the value distribution of Lerch and Selberg zeta-functions and computer calculations performed using the computational software program MATHEMATICA. The dissertation consists of the introduction, 3 chapters, the conclusions and the references. The results of the thesis are published in three scientific articles in Lithuanian and foreign journals, reported in scientific conferences in Lithuania and abroad and at the seminars of the department. In the first chapter, the limit theorems for several cases of the Lerch zeta-functions are proved. In the 1940s, Selberg proved that suitably normalized logarithm of modulus of the Riemann zeta-function on the critical line has a standard normal distribution. Selberg's proof was based on the Euler product; however, in general, Lerch zeta-functions have no Euler product. In the second chapter, the theorem concerning the zero distribution of the Lerch transendent function is proved, and computer calculations of zeros in the region Re(s)>1 are performed using MATHEMATICA. In the third chapter, the monotonicity properties of Selberg zeta-functions are investigated. Monotonicity of these two functions is directly related to the location of zeros in the critical strip. The results are compared to the monotonicity... [to full text]

The doctoral dissertation contains the material of scientific investigations done in 2008-2012 in the Faculty of Mathematics and Informatics at Vilnius University. The dissertation includes new theorems for the value distribution of Lerch and Selberg zeta-functions and computer calculations performed using the computational software program MATHEMATICA. The dissertation consists of the introduction, 3 chapters, the conclusions and the references. The results of the thesis are published in three scientific articles in Lithuanian and foreign journals, reported in scientific conferences in Lithuania and abroad and at the seminars of the department. In the first chapter, the limit theorems for several cases of the Lerch zeta-functions are proved. In the 1940s, Selberg proved that suitably normalized logarithm of modulus of the Riemann zeta-function on the critical line has a standard normal distribution. Selberg's proof was based on the Euler product; however, in general, Lerch zeta-functions have no Euler product. In the second chapter, the theorem concerning the zero distribution of the Lerch transendent function is proved, and computer calculations of zeros in the region Re(s)>1 are performed using MATHEMATICA. In the third chapter, the monotonicity properties of Selberg zeta-functions are investigated. Monotonicity of these two functions is directly related to the location of zeros in the critical strip. The results are compared to the monotonicity... [to full text] / Disertaciją sudaro mokslinių tyrimų medžiaga, kurie atlikti 2008 -2012 metais Vilniaus universitete Matematikos ir informatikos fakultete. Disertacijoje įrodomos naujos teoremos apie Lercho ir Selbergo dzeta funkcijų reikšmių pasiskirstymą, atliekami kompiuteriniai skaičiavimai matematine programa MATHEMATICA. Disertaciją sudaro įvadas, 3 skyriai, išvados ir literatūros sąrašas. Disertacijos rezultatai atspausdinti trijuose moksliniuose straipsniuose, Lietuvos ir užsienio žurnaluose, pristatyti Lietuvoje ir užsienyje vykusiose mokslinėse konferencijose bei katedros seminarų metu. Pirmajame skyriuje įrodinėjamos ribinės teoremos Lercho dzeta funkcijai. Praėjusio šimtmečio ketvirtame dešimtmetyje Selbergas įrodė, kad tinkamai normuotas Rymano dzeta funkcijos logaritmas ant kritinės tiesės turi standartinį normalųjį pasiskirstymą. Selbergo įrodymas rėmėsi Oilerio sandauga, kuria turi Rymano dzeta funkcija, bet bendru atveju jos neturi Lercho dzeta funkcija. Antrajame skyriuje įrodoma teorema apie Lercho transcendentinės funkcijos nulių įvertį vertikaliose kompleksinės plokštumos juostose bei atliekami kompiuteriniai nulių skaičiavimai srityje Re(s)>1 programa MATHEMATICA. Trečiajame skyriuje nagrinėjamos dviejų Selbergo dzeta funkcijų monotoniškumo savybės, kurios yra tiesiogiai susijusios su šių funkcijų nulių išsidėstymu kritinėje juostoje. Monotoniškumo savybės lyginamos su Rymano dzeta funkcijos monotoniškumo savybėmis ir nulių išsidėstymu, kuris yra viena didžiausių... [toliau žr. visą tekstą]

In the thesis, problems related to the moments of the periodic zeta-function are considered. The aim of the thesis is to obtain asymptotic formulae for some analytic objects related to the periodic zeta-function. The problems are the following: 1. To prove the Atkinson-type formula with a new error term in the critical strip for the periodic zeta-function with rational parameter. 2. To prove a mean square formula for the error term in the Atkinson-type formula on the critical line for the periodic zeta-function. 3. To prove a mean square formula for the error term in the Atkinson-type formula in the critical strip for the periodic zeta-function. 4. To obtain an asymptotic formula for the fourth power moment of the periodic zeta-function. / Disertacijos tyrimo objektas yra periodinė dzeta funkcija. Mokslinė problema - šios funkcijos momentų problema. Darbo tikslas - įrodyti asimptotines formules periodinės funkcijos momentams bei kai kuriems objektams, susijusiems su šios funkcijos momentais. Darbo uždaviniai yra šie: 1. Įrodyti Atkinsono tipo formulę su korektišku liekamuoju nariu kritinėje juostoje periodinei dzeta funkcijai su racionaliuoju parametru. 2. Įrodyti Atkinsono tipo formulės periodinei dzeta funkcijai kritinėje tiesėje vidurkio formulę liekamojo nario modulio kvadratui. 3. Įrodyti Atkinsono tipo formulės periodinei dzeta funkcijai kritinėje juostoje vidurkio formulę liekamojo nario modulio kvadratui. 4. Gauti asimptotinę formulę periodinės dzeta funkcijos ketvirtajam momentui.

Disertacijos tyrimo objektas yra periodinė dzeta funkcija. Mokslinė problema - šios funkcijos momentų problema. Darbo tikslas - įrodyti asimptotines formules periodinės funkcijos momentams bei kai kuriems objektams, susijusiems su šios funkcijos momentais. Darbo uždaviniai yra šie: 1. Įrodyti Atkinsono tipo formulę su korektišku liekamuoju nariu kritinėje juostoje periodinei dzeta funkcijai su racionaliuoju parametru. 2. Įrodyti Atkinsono tipo formulės periodinei dzeta funkcijai kritinėje tiesėje vidurkio formulę liekamojo nario modulio kvadratui. 3. Įrodyti Atkinsono tipo formulės periodinei dzeta funkcijai kritinėje juostoje vidurkio formulę liekamojo nario modulio kvadratui. 4. Gauti asimptotinę formulę periodinės dzeta funkcijos ketvirtajam momentui. / In the thesis, problems related to the moments of the periodic zeta-function are considered. The aim of the thesis is to obtain asymptotic formulae for some analytic objects related to the periodic zeta-function. The problems are the following: 1. To prove the Atkinson-type formula with a new error term in the critical strip for the periodic zeta-function with rational parameter. 2. To prove a mean square formula for the error term in the Atkinson-type formula on the critical line for the periodic zeta-function. 3. To prove a mean square formula for the error term in the Atkinson-type formula in the critical strip for the periodic zeta-function. 4. To obtain an asymptotic formula for the fourth power moment of the periodic zeta-function.

This thesis presents results in two apparently disparate mathematical fields which can both be examined -- and even united -- by means of pure analysis. Fractional calculus is the study of differentiation and integration to non-integer orders. Dating back to Leibniz, this idea was considered by many great mathematical figures, and in recent decades it has been used to model many real-world systems and processes, but a full development of the mathematical theory remains incomplete. Many techniques for partial differential equations (PDEs) can be extended to fractional PDEs too. Three chapters below cover my results in this area: establishing the elliptic regularity theorem, Malgrange-Ehrenpreis theorem, and unified transform method for fractional PDEs. Each one is analogous to a known result for classical PDEs, but the proof in the general fractional scenario requires new ideas and modifications. Fractional derivatives and integrals are not uniquely defined: there are many different formulae, each of which has its own advantages and disadvantages. The most commonly used is the classical Riemann-Liouville model, but others may be preferred in different situations, and now new fractional models are being proposed and developed each year. This creates many opportunities for new research, since each time a model is proposed, its mathematical fundamentals need to be examined and developed. Two chapters below investigate some of these new models. My results on the Atangana-Baleanu model proposed in 2016 have already had a noticeable impact on research in this area. Furthermore, this model and the results concerning it can be extended to more general fractional models which also have certain desirable properties of their own. Fractional calculus and zeta functions have rarely been united in research, but one chapter below covers a new formula expressing the Lerch zeta function as a fractional derivative of an elementary function. This result could have many ramifications in both fields, which are yet to be explored fully. Zeta functions are very important in analytic number theory: the Riemann zeta function relates to the distribution of the primes, and this field contains some of the most persistent open problems in mathematics. Since 2012, novel asymptotic techniques have been applied to derive new results on the growth of the Riemann zeta function. One chapter below modifies some of these techniques to prove asymptotics to all orders for the Hurwitz zeta function. Many new ideas are required, but the end result is more elegant than the original one for Riemann zeta, because some of the new methodologies enable different parts of the argument to be presented in a more unified way. Several related problems involve asymptotics arbitrarily near a stationary point. Ideally it should be possible to find uniform asymptotics which provide a smooth transition between the integration by parts and stationary phase methods. One chapter below solves this problem for a particular integral which arises in the analysis of zeta functions.

Multifractals have during the past 20 − 25 years been the focus of enormous attention in the mathematical literature. Loosely speaking there are two main ingredients in multifractal analysis: the multifractal spectra and the Renyi dimensions. One of the main goals in multifractal analysis is to understand these two ingredients and their relationship with each other. Motivated by the powerful techniques provided by the use of the Artin-Mazur zeta-functions in number theory and the use of the Ruelle zeta-functions in dynamical systems, Lapidus and collaborators (see books by Lapidus & van Frankenhuysen [32, 33] and the references therein) have introduced and pioneered use of zeta-functions in fractal geometry. Inspired by this development, within the past 7−8 years several authors have paralleled this development by introducing zeta-functions into multifractal geometry. Our result inspired by this work will be given in section 2.2.2. There we introduce geometric multifractal zeta-functions providing precise information of very general classes of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. Results in that section are based on paper [37]. Dynamical zeta-functions have been introduced and developed by Ruelle [63, 64] and others, (see, for example, the surveys and books [3, 54, 55] and the references therein). It has been a major challenge to introduce and develop a natural and meaningful theory of dynamical multifractal zeta-functions paralleling existing theory of dynamical zeta functions. In particular, in the setting of self-conformal constructions, Olsen [49] introduced a family of dynamical multifractal zeta-functions designed to provide precise information of very general classes of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. However, recently it has been recognised that while self-conformal constructions provide a useful and important framework for studying fractal and multifractal geometry, the more general notion of graph-directed self-conformal constructions provide a substantially more flexible and useful framework, see, for example, [36] for an elaboration of this. In recognition of this viewpoint, in section 2.3.11 we provide main definitions of the multifractal pressure and the multifractal dynamical zeta-functions and we state our main results. This section is based on paper [38]. Setting we are working unifies various different multifractal spectra including fine multifractal spectra of self-conformal measures or Birkhoff averages of continuous function. It was introduced by Olsen in [43]. In section 2.1 we propose answer to problem of defining Renyi spectra in more general settings and provide slight improvement of result regrading multifractal spectra in the case of Subshift of finite type.

In the first part of this thesis we transfer a result of Guillopé et al. concerning the number of zeros of the Selberg zeta function for convex cocompact Schottky groups to the setting of certain types of graph directed Markov systems (GDMS). For these systems the zeta function will be a type of Ruelle zeta function. We show that for a finitely generated primitive conformal GDMS S, which satisfies the strong separation condition (SSC) and the nestedness condition (NC), we have for each c>0 that the following holds, for each w \in\$C$ with Re(w)>-c, |\Im(w)|>1 and for all k \in\$N$ sufficiently large: log | zeta(w) | <<e {delta(S).log(Im|w|)} and card{w \in\ Q(k) | zeta(w)=0} << k {delta(S)}. Here, Q(k)\subset\%C$ denotes a certain box of height k, and delta(S) refers to the Hausdorff dimension of the limit set of S. In the second part of this thesis we show that in any dimension m \in\$N$ there are GDMSs for which the Hausdorff dimension of the uniformly radial limit set is equal to a given arbitrary number d \in\(0,m) and the Hausdorff dimension of the Jørgensen limit set is equal to a given arbitrary number j \in\ [0,m). Furthermore, we derive various relations between the exponents of convergence and the Hausdorff dimensions of certain different types of limit sets for iterated function systems (IFS), GDMSs, pseudo GDMSs and normal subsystems of finitely generated GDMSs. Finally, we apply our results to Kleinian groups and generalise a result of Patterson by showing that in any dimension m \in\$N$ there are Kleinian groups for which the Hausdorff dimension of their uniformly radial limit set is less than a given arbitrary number d \in\ (0,m) and the Hausdorff dimension of their Jørgensen limit set is equal to a given arbitrary number j \in\ [0,m).