No description available.

In 1997, Xian-Jin Li gave an equivalence to the classical Riemann hypothesis, now referred to as Li's criterion, in terms of the non-negativity of a particular infinite sequence of real numbers. We formulate the analogue of Li's criterion as an equivalence for the generalized quasi-Riemann hypothesis for functions in an extension of the Selberg class, and give arithmetic formulae for the corresponding Li coefficients in terms of parameters of the function in question. Moreover, we give explicit non-negative bounds for certain sums of special values of polygamma functions, involved in the arithmetic formulae for these Li coefficients, for a wide class of functions. Finally, we discuss an existing result on correspondences between zero-free regions and the non-negativity of the real parts of finitely many Li coefficients. This discussion involves identifying some errors in the original source work which seem to render one of its theorems conjectural. Under an appropriate conjecture, we give a generalization of the result in question to the case of Li coefficients corresponding to the generalized quasi-Riemann hypothesis. We also give a substantial discussion of research on Li's criterion since its inception, and some additional new supplementary results, in the first chapter. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-07-31 13:14:03.414

Universiẗat, Diss., 2001--Mannheim.

No description available.

Em teoria dos números, o teorema de Erdös-Kac, também conhecido como o teorema fundamental de teoria probabilística dos números, diz que se w(n) denota a quantidade de fatores primos distintos de n, então a sequência de funções de distribuições N definidas por FN(x) = (1/N) #{n <= N : (w(n) log log N)/(log log N)^(1/2)} <= x}, converge uniformemente sobre R para a distribuição normal padrão. Neste trabalho desenvolvemos todos os teoremas necessários para uma demonstração analítica, que nos permitirá encontrar a ordem de erro da convergência acima. / In number theory, the Erdös-Kac theorem, also known as the fundamental theorem of probabilistic number theory, states that if w(n) is the number of distinct prime factors of n, then the sequence of distribution functions N, defined by FN(x) = (1/N) #{n <= N : (w(n) log log N)/(log log N)^(1/2)} <= x}, converges uniformly on R to the standard normal distribution. In this work we developed all theorems needed to an analytic demonstration, which will allow us to find an order of error of the above convergence.

Em teoria dos números, o teorema de Erdös-Kac, também conhecido como o teorema fundamental de teoria probabilística dos números, diz que se w(n) denota a quantidade de fatores primos distintos de n, então a sequência de funções de distribuições N definidas por FN(x) = (1/N) #{n <= N : (w(n) log log N)/(log log N)^(1/2)} <= x}, converge uniformemente sobre R para a distribuição normal padrão. Neste trabalho desenvolvemos todos os teoremas necessários para uma demonstração analítica, que nos permitirá encontrar a ordem de erro da convergência acima. / In number theory, the Erdös-Kac theorem, also known as the fundamental theorem of probabilistic number theory, states that if w(n) is the number of distinct prime factors of n, then the sequence of distribution functions N, defined by FN(x) = (1/N) #{n <= N : (w(n) log log N)/(log log N)^(1/2)} <= x}, converges uniformly on R to the standard normal distribution. In this work we developed all theorems needed to an analytic demonstration, which will allow us to find an order of error of the above convergence.

<p>This thesis in analytic number theory consists of 3 parts and 13 individual papers.</p><p>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.</p><p>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.</p><p>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.</p><p>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.</p><p>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.</p>

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.

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]