Dissertations / Theses on the topic 'Bernoulli number'

En premiere partie, on donne des congruences entre nombres de bernoulli-hcowitz dans le cas supersingulier. En deuxieme partie, on montre que la suite des nombres de bernoulli verifie des formules de recurrence qui servent a tester si un nombre premier est irregulier. En troisieme partie, on etudie les zeros des polynomes de bessel generalises, en particulier on encadre un zero reel, apres developpement asymptotique, et on donne des estimations uniformes des valeurs de ces polynomes

Un entier naturel est dit $y$-friable lorsque son plus grand facteur premier n'excède pas $y$. Ce travail est consacré à l'étude des entiers friables dans le cadre de la théorie analytique et probabiliste des nombres. La première partie est dévolue à un problème posé par Davenport en 1937, qui consiste à déterminer les conditions de validité de diverses généralisations de son développement de la fonction sinus en série de parties fractionnaires. Ces généralisations peuvent être décrites par un couple de fonctions arithmétiques, liées par la relation de convolution $f=g*\1$. Nous traitons le cas où $g$ est la fonction de Piltz d'ordre $z\in\CC$. La deuxième partie est consacrée à l'étude du comportement asymptotique de la constante optimale dans une version friable de l'inégalité de Turán-Kubilius. Précisant des résultats récents de La Bretèche et Tenenbaum, nous généralisons au cas friable une formule asymptotique de la variance d'une fonction arithmétique additive, établie par Hildebrand en 1983.

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Neste trabalho,estudamos os números e os polinomios de Bernoulli,bem como algumas de suas aplicações mais importantes em Teoria dos Números. Com base em uma caracterização ao simples, os polinômios de Bernoulli são introduzidos e, posteriormente, os números de Bernoulli. As séries de Fourier dos polinomios de Bernoulli são utilizadas na demonstração da equação funcional da função teta. Esta equação, por sua vez, é utilizada na demonstração da celebre equação funcional da função zeta, que tem importância central na teoria da distribuição dos números primos. Além das conexões com a funções especiais zeta e teta, discutimos também, em detalhe,conexões entre os números e os polinomios de Bernoulli com a função gama. Essas relações são então exploradas para produzir belas fórmulas para certos valores da função zeta, entre outras aplicações.
In this work we study Bernoulli numbers and Bernoulli polynomials, as well as some of its most important applications in Number Theory. Based on a simple characterization, the Bernoulli polynomials are introduced and, later, the Bernoulli numbers. The Fourier series of the Bernoulli polynomials are used to demonstrate the functional equation of the theta function. This equation, in turn, is used in the proof of the famous functional equation of the zeta function, which is central to the theory of prime number distribution. In addition to the connections with the special functions zeta and theta, we also discuss, in detail, connections between the Bernoulli numbers and Bernoulli polynomials with the gamma function. These relations are then explored to produce beautiful formulas for certain values of the zeta function,among other applications.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
In this work some topics of the Discrete or Finite Calculus are developed. In particular, we study difference operators, factorial powers, Stirling numbers of the first and second type, the Newton’s formula of differences, the fundamental theorem of the Finite Calculus, the summation process, and the Bernoulli numbers and Bernoulli polynomials. Then we show the effectiveness of the theory for the calculation of closed formulas for the value of many finite sums. We also study the classical problem of obtaining the polynomials which express the value of the sums of powers of natural numbers.
Neste trabalho desenvolvemos alguns tópicos do Cálculo Discreto ou Finito. Em particular, estudamos operadores de diferenças, potências fatoriais, números de Stirling do primeiro e do segundo tipo, a fórmula de diferenças de Newton, o teorema fundamental do Cálculo Finito, o processo de somação e os números e polinômios de Bernoulli. Mostramos então a eficácia da teoria no cálculo de fórmulas fechadas para o valor de diversas somas finitas. Também estudamos o problema clássico de obter os polinômios que expressam o valor de somas de potências de números naturais.

碩士
國立臺中教育大學
數學教育學系
96
Up to the present, it is an important study for calculating Bernoulli number. There are many different methods to claculate Bernoulli number. But for these methods, we must take lots of steps to calaulate Bernoulli number. Based on this, our research applies Riemann--zeta function and the extended function of the sums of powers of consecutive integers to get an easier method. Then, we will calculate Bernoulli number by using Matlab 7.1, and investigate the relationship between Bernoulli nmuber and Stirling number of second kind. Our results are as follows. 1. The formula of Bernoulli number is B_{2k}=\frac{1}{2k+1} \left \{ C_{2k}^{2k+1}S_{1}^{\prime}(-1) + \sum_{i=1}^{k}C_{2i+1}^{2k+1} S_{2k-2i}^{\prime}(-1) \right \}, k\in N . 2. When $k$ is bigger, Bernoulli number will become bigger and be alternated between plus and minus. 3. The relationship between Bernoulli number and Stirling number of second kind is B_{m+1}=\sum_{k=1}^{m+1}\frac{(-1)^k}{k+1}\cdot k!\cdot S_2(m+1,k).

碩士
國立臺中教育大學
數學教育學系
96
This research hung over from the extended functions for the sum of powers of consecutive integers, we colleted the literatures of the related research about the functions of Riemann zeta and Bernoulli Number, both newly interpreted and predigested the properties of the functions of Riemann zeta and Bernoulli Number. Thus we built the connection between the functions of Riemann zeta and Bernoulli Number, according to \zeta(2 k)=(-1)^{k-1} 2^{2k-1} \frac{B_{2k} \pi^{2k}}{(2k)!}, \ k \in \mathbb{N},and S_{2k}^{\prime}(-1)=\frac{(-1)^{k-1} (2k)!}{2^{2k-1} (\pi)^{2k}}\zeta(2k), S_{2k+1}^{\prime}(-1)=0,Take the function of Riemann zeta as bridge, we find that S_{2k}^{\prime}(-1)=B_{2k},B_{2k}=\frac{1}{2k+1} \left \{ C_{2k}^{2k+1} S_{1}^{\prime}(-1)+ \sum_{i=1}^{k} C_{2i+1}^{2k+1} S_{2k-2i}^{\prime}(-1) \right \},where $S_k^{\prime}(x)$ denotes the first derivative of $S_k(x)$ for each positive integer $k$.

Presents a method of deriving the limiting distributions of the number of occurences of success (S) runs of length k for all types of runs under the Markovian structure with stationary transition probabilities.

碩士
國立中正大學
數學研究所
89
In this paper, we will first introduce some basic properties in Bernoulli numbers (resp. polynomials) and Euler numbers (resp. polynomials), and then prove that the special values of certain zeta functions at non-positive integers can be expressed by Euler polynomials. However, there are usually more than one way to express the special values at non-positive integers. This leads to classical identities as well as new identities among Euler numbers and Euler polynomials. And, we can also derive some new identities among Bernoulli numbers (resp. polynomials) and Euler numbers (resp. polynomials).

Kwok, Tsz Chiu.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.
Includes bibliographical references (leaves 43-45).
Abstracts in English and Chinese.
Chapter 1 --- Introduction --- p.6
Chapter 1.1 --- Historical remarks and main results --- p.6
Chapter 1.2 --- Structure of the thesis --- p.8
Chapter 2 --- Basic properties --- p.10
Chapter 2.1 --- Existence of infinite convolution --- p.10
Chapter 2.2 --- Properties --- p.16
Chapter 2.3 --- Law of pure type --- p.17
Chapter 3 --- Some results related to pure singularity --- p.20
Chapter 3.1 --- The Pisot-Vijayaraghavan numbers --- p.20
Chapter 3.2 --- The Salem numbers --- p.22
Chapter 3.3 --- The weak separation condition --- p.23
Chapter 4 --- A proof of almost everywhere absolute continuity --- p.30
Chapter 5 --- Other results and problems --- p.37
Chapter 5.1 --- Entropy of Bernoulli convolutions --- p.37
Chapter 5.2 --- Dimensions --- p.40
Chapter 5.3 --- Non PV numbers with bad behavior --- p.41
Chapter 5.4 --- Open problems --- p.41
Bibliography --- p.43