<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.

Made available in DSpace on 2014-12-17T14:55:01Z (GMT). No. of bitstreams: 1 MarcosADA.pdf: 1444489 bytes, checksum: 289536fadcf88cdfafb2eefa6b4f2ac4 (MD5) Previous issue date: 2003-12-29 / Most algorithms for state estimation based on the classical model are just adequate for use in transmission networks. Few algorithms were developed specifically for distribution systems, probably because of the little amount of data available in real time. Most overhead feeders possess just current and voltage measurements at the middle voltage bus-bar at the substation. In this way, classical algorithms are of difficult implementation, even considering off-line acquired data as pseudo-measurements. However, the necessity of automating the operation of distribution networks, mainly in regard to the selectivity of protection systems, as well to implement possibilities of load transfer maneuvers, is changing the network planning policy. In this way, some equipments incorporating telemetry and command modules have been installed in order to improve operational features, and so increasing the amount of measurement data available in real-time in the System Operation Center (SOC). This encourages the development of a state estimator model, involving real-time information and pseudo-measurements of loads, that are built from typical power factors and utilization factors (demand factors) of distribution transformers. This work reports about the development of a new state estimation method, specific for radial distribution systems. The main algorithm of the method is based on the power summation load flow. The estimation is carried out piecewise, section by section of the feeder, going from the substation to the terminal nodes. For each section, a measurement model is built, resulting in a nonlinear overdetermined equations set, whose solution is achieved by the Gaussian normal equation. The estimated variables of a section are used as pseudo-measurements for the next section. In general, a measurement set for a generic section consists of pseudo-measurements of power flows and nodal voltages obtained from the previous section or measurements in real-time, if they exist -, besides pseudomeasurements of injected powers for the power summations, whose functions are the load flow equations, assuming that the network can be represented by its single-phase equivalent. The great advantage of the algorithm is its simplicity and low computational effort. Moreover, the algorithm is very efficient, in regard to the accuracy of the estimated values. Besides the power summation state estimator, this work shows how other algorithms could be adapted to provide state estimation of middle voltage substations and networks, namely Schweppes method and an algorithm based on current proportionality, that is usually adopted for network planning tasks. Both estimators were implemented not only as alternatives for the proposed method, but also looking for getting results that give support for its validation. Once in most cases no power measurement is performed at beginning of the feeder and this is required for implementing the power summation estimations method, a new algorithm for estimating the network variables at the middle voltage bus-bar was also developed / A grande maioria dos algoritmos de estima??o de estado, que usa o modelo cl?ssico, se destina ? aplica??o em sistemas de transmiss?o. H? poucos algoritmos para sistemas de distribui??o. Isto se deve em parte, a pequena quantidade de dados de medi??o dispon?veis em tempo real. A maioria dos alimentadores s? disp?e de medi??o de corrente na sa?da do barramento de m?dia tens?o da subesta??o. Dessa forma, a aplica??o de algoritmos tradicionais de estima??o de estado para a supervis?o de alimentadores pode ser inadequada, mesmo considerando dados obtidos off-line atrav?s de pseudomedi??es. Entretanto, a necessidade de automatiza??o da opera??o dos sistemas de distribui??o, principalmente no que diz respeito ? seletividade quando da presen?a de defeitos, fez surgir alguns equipamentos telecomandados, que incorporam m?dulos de telemedi??o de algumas grandezas da rede, que podem ser transmitidas em tempo real para o centro de opera??o do sistema COS. Isso permite o desenvolvimento de um novo modelo de estimador de estado, envolvendo medidas reais e pseudomedidas de cargas, que s?o constru?das a partir da defini??o de fatores de pot?ncia e de utiliza??o t?picos de sistemas de distribui??o. O presente trabalho trata do desenvolvimento de um novo modelo de estimador de estado voltado para sistemas de distribui??o, particularmente, alimentadores radiais. Baseia-se no algoritmo do fluxo de carga soma de pot?ncias. Da? o nome estimador de estado de soma de pot?ncias. O m?todo faz a estima??o de alimentador por se??o, partindo da subesta??o para os ramais. Para cada se??o ? constru?do o modelo de medi??o. Isto resulta em sistemas de equa??es n?o-lineares, sobre determinados, que requerem uma solu??o iterativa. Obt?m-se essa solu??o atrav?s do m?todo dos m?nimos quadrados ponderados via equa??o normal de Gauss. As grandezas estimadas em uma se??o s?o usadas como pseudomedidas para estimar a se??o subseq?ente. O conjunto de medi??o de cada se??o ? constitu?do por pseudomedidas ou medidas de fluxos de pot?ncia nos trechos e tens?es nodais, em tempo real, e por pseudomedidas de inje??es de pot?ncias nos n?s. As pseudomedidas de inje??es de pot?ncia s?o constru?das a partir das equa??es cl?ssicas de pot?ncias injetadas, usadas no estudo de fluxo de carga. Assume-se ainda, que o sistema trif?sico pode ser representado por seu equivalente monof?sico. A grande vantagem do algoritmo est? na simplicidade e rapidez do programa computacional que o implementa. Al?m disso, ? muito eficiente no que diz respeito ? exatid?o das grandezas estimadas. Al?m do estimador soma de pot?ncias, este trabalho mostra como outros algoritmos poderiam ser adaptados para prover estima??o de estado de subesta??es e circuitos de m?dia tens?o, isto ?, o m?todo de Schweppe e um algoritmo baseado em proporcionalidade de corrente, que normalmente ? usado em estudos de planejamento de redes. Ambos os estimadores foram implementados n?o somente como alternativas para o m?todo proposto, mas tamb?m procurando obter resultados para servir de suporte para sua valida??o. Uma vez que na maioria dos casos n?o h? medi??o de pot?ncias na sa?da para o alimentador e esta ? requerida para implementa??o do m?todo da soma de pot?ncias, um novo algoritmo para estimar as grandezas de rede em barra de m?dia tens?o foi tamb?m desenvolvido