DOCSLIB.ORG

An Introduction to the Riemann Hypothesis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Introduction to the Riemann Hypothesis An introduction to the Riemann hypothesis Author: Alexander Bielik [email protected] Supervisor: P¨arKurlberg SA104X { Degree Project in Engineering Physics Royal Institute of Technology (KTH) Department of Mathematics September 13, 2014 Abstract This paper exhibits the intertwinement between the prime numbers and the zeros of the Riemann zeta function, drawing upon existing literature by Davenport, Ahlfors, et al. We begin with the meromorphic continuation of the Riemann zeta function ζ and the gamma function Γ. We then derive a functional equation that relates these functions and formulate the Riemann hypothesis. We move on to the topic of finite-ordered functions and their Hadamard products. We show that the xi function ξ is of finite order, whence we obtain many useful properties. We then use these properties to find a zero-free region for ζ in the critical strip. We also determine the vertical distribution of the non-trivial zeros. We finally use Perron's formula to derive von Mangoldt's explicit formula, which is an approximation of the Cheby- shev function . Using this approximation, we prove the prime number theorem and conclude with an implication of the Riemann hypothesis. Contents Introduction 2 1 The statement of the Riemann hypothesis3 1.1 The Riemann zeta function ζ .........................................3 1.2 The gamma function Γ.............................................4 1.3 The functional equation............................................7 1.4 The critical strip................................................8 2 Zeros in the critical strip 10 2.1 Functions of finite order............................................ 10 2.2 The Hadamard product for functions of order 1............................... 11 2.3 Proving that ξ has order at most 1...................................... 13 2.4 A zero-free region for ζ ............................................. 16 2.5 The number of zeros in a rectangle...................................... 19 3 The distribution of prime numbers 23 3.1 Perron's formula for Dirichlet series...................................... 23 3.2 An approximation of the Chebyshev function ............................... 25 3.3 von Mangoldt's explicit formula........................................ 27 3.4 Proving the explicit formula.......................................... 28 3.5 The prime number theorem.......................................... 30 3.6 The smallest possible error term....................................... 32 Appendices 33 A Additional proofs 34 A.1 The Weierstrass form of Γ........................................... 34 A.2 Stirling's formula for ln Γ........................................... 35 A.3 Jensen's formula for holomorphic functions................................. 37 A.4 The uniqueness theorem for Dirichlet series................................. 38 Bibliography 39 1 Introduction In 1859, the German mathematician Georg Friedrich Bernhard Riemann proposed a hypothesis [Riemann, 1859, pp. 1-9] about prime numbers that would later bear his name, the Riemann hypothesis. The prime numbers do not appear to follow any obvious pattern. However, Riemann observed a close relation between the behavior of an elaborate function, the so-called Riemann zeta function, and the frequency of prime numbers. Riemann calculated a few zeros of this function, and quickly noted that the interesting ones lay on a certain vertical straight line in the complex plane. Riemann subsequently conjectured that all non-trivial zeros lie on this line. Today, over 1013 zeros are known [Gourdon, 2004, pp. 19-25], and all of them agree with the hypothesis. The Riemann hypothesis has important implications for the distribution of prime numbers and is strongly con- nected to the prime number theorem, which gives a good approximation of the density of prime numbers. In particular, the Riemann hypothesis gives a precise answer to how good the approximation given by the prime number theorem is. In a sense, the Riemann hypothesis conveys the idea that the prime numbers are distributed as regularly as possible. This regularity would tell a great deal about the average behavior of prime numbers in the long run. In today's society, the importance of prime numbers has increased rapidly, especially with the advance of in- formation technology and cryptography. However, the importance of the Riemann hypothesis goes far beyond its consequences for the distribution of prime numbers. It has been shown that hundreds of statements in number theory follow from it [Gowers et al., 2008, p. 715]. With this background, it might not be a surprise that some mathemati- cians consider the hypothesis to be the most important problem in pure mathematics, but it remains unresolved for now. The Riemann hypothesis is one of the seven Millennium Prize Problems that were stated by the Clay Mathe- matics Institute in 2000, carrying a million dollar prize for a correct solution. It is also part of the eighth problem in David Hilbert's list of unsolved problems. Other than the Riemann zeta function and its zeroes, there is currently no known approach to establish the distri- bution of prime numbers with desired precision. A disproof of the Riemann hypothesis would reveal a lot about how disordered the primes numbers really are. Enrico Bombieri, a prominent number theorist, remarked that "the failure of the Riemann hypothesis would create havoc in the distribution of prime numbers" [Havil, 2003, p. 205]. In 1770, Euler argued more pessimistically that "mathematicians have tried in vain to discover some order in the sequence of prime numbers but we have every reason to believe that there are some mysteries which the human mind will never penetrate" [Gowers et al., 2008, p. 348]. 2 1. The statement of the Riemann hypothesis 1.1 The Riemann zeta function ζ For complex numbers s with real part greater than 1, we define the Riemann zeta function by the absolutely convergent series 1 X 1 1 1 ζ(s) := = 1 + + + ··· : (1.1) ns 2s 3s n=1 By the Weierstrass M-test, we find that the convergence is uniform in the region Re(s) ≥ 1 + δ for any δ > 0. We now show that ζ is holomorphic within Re(s) > 1. To this end, we consider the sequence of holomorphic functions defined by i X 1 f (s) := : (1.2) i ns n=1 1 Since ffigi=1 converges uniformly to ζ on any compact subset of Re(s) > 1, it follows that the function is holomorphic there. In 1737, Euler deduced that 1 n Y 1 Y X 1 = 1 − p−s ps p2P p2P n=0 Y 1 1 = 1 + + + ··· ps p2s p2P X 1 X 1 = 1 + + + ··· ps psqs p2P p;q2P 1 1 1 X 1 = 1 + + + ··· = = ζ(s) (1.3) 2s 3s ns n=1 for any integer s > 1, where P denotes the set of prime numbers, though his argument can be extended to any complex number s with Re(s) > 1. In the above derivation, we first used the formula for a geometric series. Then, we rewrote the product and used the fundamental theorem of arithmetic, which states that each positive integer equals exactly one product of prime powers. This useful relation is usually called the Euler product formula. Some writers describe it as "the Golden Key" [Derbyshire, 2004, p. 105]. Taking the logarithm, we find that Y 1 X 1 ln = ln 1 − p−s 1 − p−s p2P p2P s X p = ln ps − 1 p2P X s s = ln jp j − ln jp − 1j p2P jpsj jpsj X dx X dx X 1 = ≤ < s ; (1.4) ˆjps−1j x ˆjps|−1 x jp j − 1 p2P p2P p2P 3 which clearly converges for Re(s) > 1. It follows that the product on the left-hand side of (1.3) converges. Hence, the formula (1.3) allows us to conclude that ζ has no zeros in the region Re(s) > 1, as each factor in the convergent product is different from zero. By partial summation, we can extend the domain of the function to Re(s) > 0: 1 X ζ(s) = n−s n=1 1 X = n n−s − (n + 1)−s n=1 1 X n+1 = s n x−s−1 dx ˆ n=1 n 1 = s bxcx−s−1 dx ˆ1 1 1 = s x−s dx − s fxgx−s−1 dx ˆ1 ˆ1 s 1 = − s fxgx−s−1 dx: (1.5) s − 1 ˆ1 Here, the symbols bxc and fxg = x − bxc stand for the integral and fractional parts of x, respectively. Since jfxgj ≤ 1, we see that the integral on the right of (1.5) converges absolutely in this extended domain. Furthermore, the convergence is uniform in the region Re(s) ≥ δ for any δ > 0. It follows that this new function is a meromorphic continuation of ζ. We observe that its only pole in this domain is a simple pole at s = 1 with residue Res(ζ; 1) = lim(s − 1)ζ(s) = 1: (1.6) s!1 We also observe that both terms in (1.5) are real and negative for real numbers 0 < s < 1, so ζ(s) < 0 on this line. Consequently, there are no zeros in this interval. For completeness, we mention another way to meromorphically continue ζ to Re(s) > 0. For Re(s) > 1, we observe that 1 1 1 2 X 1 X 2 X (−1)n−1 1 − ζ(s) = − = =: η(s); (1.7) 2s ns (2n)s ns n=1 n=1 n=1 whence ζ(s) = (1 − 21−s)−1η(s). Here, η is the Dirichlet eta function, which can be shown to converge for Re(s) > 0. 2πi While the right-hand side has singularities at s = 1+k ln 2 , where k 2 Z, only the singularity at s = 1 is non-removable. 1.2 The gamma function Γ The gamma function is an extension of the factorial function to complex numbers. For numbers s with positive real part, it is defined by the convergent integral 1 Γ(s) := ts−1e−t dt: (1.8) ˆ0 Let us show that this function is holomorphic. This time, we use Morera's theorem.
Load more

Recommended publications
  • GPU-Based Visualization of Domain-Coloured Algebraic Riemann Surfaces
    GPU-based visualization of domain-coloured algebraic Riemann surfaces Stefan Kranich∗ 11 November 2015 Abstract We examine an algorithm for the visualization of domain-coloured Riemann surfaces of plane algebraic curves. The approach faithfully reproduces the topology and the holomorphic structure of the Riemann surface. We discuss how the algorithm can be implemented efficiently in OpenGL with geometry shaders, and (less efficiently) even in WebGL with multiple render targets and floating point textures. While the generation of the surface takes noticeable time in both implementations, the visualization of a cached Riemann surface mesh is possible with interactive performance. This allows us to visually explore otherwise almost unimaginable mathematical objects. As examples, we look at the complex square root and the folium of Descartes. For the folium of Descartes, the visualization reveals features of the algebraic curve that are not obvious from its equation. 1 Introduction 1.1 Mathematical background The following basic example illustrates what we would like to visualize. Example 1.1. Let y be the square root of x, p y = x: If x is a non-negative real number, we typically define y as the non-negative real arXiv:1507.04571v3 [cs.GR] 10 Nov 2015 number whose square equals x, i.e. we always choose the non-negative solution of the equation y2 − x = 0 (1) p as y = x. For negative real numbers x, no real number y solves Equation 1. However, if we define the imaginary unit i as a number with the property that i2 = −1 then the square root of x becomes the purely imaginary number y = ipjxj: ∗Zentrum Mathematik (M10), Technische Universit¨atM¨unchen, 85747 Garching, Germany; E-mail address: [email protected] 1 Im Im Re Re Figure 1.2: When a complex number (black points) runs along a circle centred at the origin of the complex plane, its square roots (white points) move at half the angular velocity (left image).
    [Show full text]
  • Visualization of Complex Function Graphs in Augmented Reality
    M A G I S T E R A R B E I T Visualization of Complex Function Graphs in Augmented Reality ausgeführt am Institut für Softwaretechnik und Interaktive Systeme der Technischen Universität Wien unter der Anleitung von Univ.Ass. Mag. Dr. Hannes Kaufmann durch Robert Liebo Brahmsplatz 7/11 1040 Wien _________ ____________________________ Datum Unterschrift Abstract Understanding the properties of a function over complex numbers can be much more difficult than with a function over real numbers. This work provides one approach in the area of visualization and augmented reality to gain insight into these properties. The applied visualization techniques use the full palette of a 3D scene graph's basic elements, the complex function can be seen and understood through the location, the shape, the color and even the animation of a resulting visual object. The proper usage of these visual mappings provides an intuitive graphical representation of the function graph and reveals the important features of a specific function. Augmented reality (AR) combines the real world with virtual objects generated by a computer. Using multi user AR for mathematical visualization enables sophisticated educational solutions for studies dealing with complex functions. A software framework that has been implemented will be explained in detail, it is tailored to provide an optimal solution for complex function graph visualization, but shows as well an approach to visualize general data sets with more than 3 dimensions. The framework can be used in a variety of environments, a desktop setup and an immersive setup will be shown as examples. Finally some common tasks involving complex functions will be shown in connection with this framework as example usage possibilities.
    [Show full text]
  • Domain Coloring of Complex Functions
    Domain Coloring of Complex Functions Contents 1 Introduction Not only in mathematic we are confronted with graphs of all possible data sets. In economics and science we want to present the results by plotting their function graph inside a optimal coordinate system. In complex analysis we work with holomorphic functions which are complex differen- tiable. Like in real analysis we are interesting to plot such functions as graph inside a optimal coordinate systems. The graph of holomorphic functions is a subset of the four-dimensional coordinate system. But we are limited to three dimension, because we do not know how to draw objects in higher dimensional spaces. Therefore we are forced to construct a methode to visualize such functions. In this sec- tion we give a short overview of the domain coloring which allows us to visualize the graph of complex funtions by using colors. First we give a overview of complex analysis and holomorphic function. After we have presented the main idea behind the domain coloring we discuss a example to understand the main concept of domain coloring. 1 2 Introduction to Complex Analysis 2 Introduction to Complex Analysis In this section we give a short review of the main idea of complex analysis which we need to understand the methode of domain coloring. The reader should familiar with the construction of complex numbers and the representation of such numbers with polar coordinates. In complex anlysis we work with complex function which consist of three parts: first, a set D ⊂ C of input values, which is called the domain of the function, second, the range of f in C and third, for every input value x 2 D, a unique function value f(x) in the range of f.
    [Show full text]
  • Domain Coloring of Complex Functions
    Domain Coloring of Complex Functions Konstantin Poelke and Konrad Polthier 1 Introduction 2 What is a Function? Let us briefly recap the definition of a Visualizing functions is an omnipresent function to fix terminology. A function f task in many sciences and almost every day consists of three parts: first, a set D of in- we are confronted with diagrams in news- put values, which is called the domain of the papers and magazines showing functions of function, second, a set Y called the range of all possible flavours. Usually such func- f and third, for every input value x ∈ D, tions are visualized by plotting their func- a unique value y ∈ Y , called the function tion graph inside an appropriate coordinate value of f at x, denoted f(x). The set Γ(f) system, with the probably most prominent of all pairs (a, f(a)), a ∈ D, is a subset of choice being the cartesian coordinate sys- the product set D × Y and called the func- tem. This allows us to get an overall im- tion graph of f. pression of the function’s behaviour as well One particular type of functions that are as to detect certain distinctive features such widely used in engineering and physics are as minimal or maximal points or points complex functions, i.e. functions f : D ⊆ where the direction of curvature changes. C → Y ⊆ C whose domain and range In particular, we can “see” the dependence are subsets of the complex numbers, and between input and output. However, this we will focus on complex functions in the technique is limited to three dimensions, following.
    [Show full text]
  • Multiplication Modulo N Along the Primorials with Its Differences And
    Multiplication Modulo n Along The Primorials With Its Differences And Variations Applied To The Study Of The Distributions Of Prime Number Gaps A.K.A. Introduction To The S Model Russell Letkeman r. letkeman@ gmail. com Dedicated to my son Panha May 19, 2013 Abstract The sequence of sets of Zn on multiplication where n is a primorial gives us a surprisingly simple and elegant tool to investigate many properties of the prime numbers and their distributions through analysis of their gaps. A natural reason to study multiplication on these boundaries is a construction exists which evolves these sets from one primorial boundary to the next, via the sieve of Eratosthenes, giving us Just In Time prime sieving. To this we add a parallel study of gap sets of various lengths and their evolution all of which together informs what we call the S model. We show by construction there exists for each prime number P a local finite probability distribution and it is surprisingly well behaved. That is we show the vacuum; ie the gaps, has deep structure. We use this framework to prove conjectured distributional properties of the prime numbers by Legendre, Hardy and Littlewood and others. We also demonstrate a novel proof of the Green-Tao theorem. Furthermore we prove the Riemann hypoth- esis and show the results are perhaps surprising. We go on to use the S model to predict novel structure within the prime gaps which leads to a new Chebyshev type bias we honorifically name the Chebyshev gap bias. We also probe deeper behavior of the distribution of prime numbers via ultra long scale oscillations about the scale of numbers known as Skewes numbers∗.
    [Show full text]
  • Una Introducción Al Método De Dominio Colorado Con Geogebra Para
    101 Una introducción al método de dominio colorado con GeoGebra para la visualización y estudio de funciones complejas Uma introdução ao método do domínio colorado com GeoGebra para visualizar e estudar funções complexas An introduction the method domain coloring with GeoGebra for visualizing and studying complex JUAN CARLOS PONCE CAMPUZANO1 0000-0003-4402-1332 researchgate.net/profile/Juan_Ponce_Campuzano geogebra.org/u/jcponce http://dx.doi.org/10.23925/2237-9657.2020.v9i1p101-119 RESUMEN Existen diversos métodos para visualizar funciones complejas, tales como graficar por separado sus componentes reales e imaginarios, mapear o transformar una región, el método de superficies analíticas y el método de dominio coloreado. Este último es uno de los métodos más recientes y aprovecha ciertas características del color y su procesamiento digital. La idea básica es usar colores, luminosidad y sombras como dimensiones adicionales, y para visualizar números complejos se usa una función real que asocia a cada número complejo un color determinado. El plano complejo puede entonces visualizarse como una paleta de colores construida a partir del esquema HSV (del inglés Hue, Saturation, Value – Matiz, Saturación, Valor). Como resultado, el método de dominio coloreado permite visualizar ceros y polos de funciones, ramas de funciones multivaluadas, el comportamiento de singularidades aisladas, entre otras propiedades. Debido a las características de GeoGebra en cuanto a los colores dinámicos, es posible implementar en el software el método de dominio coloreado para visualizar y estudiar funciones complejas, lo cual se explica en detalle en el presente artículo. Palabras claves: funciones complejas, método de dominio coloreado, colores dinámicos. RESUMO Existem vários métodos para visualizar funções complexas, como plotar seus componentes reais e imaginários separadamente, mapear ou transformar uma região, o método de superfície analítica e o método de domínio colorido.
    [Show full text]
  • Orthogonal Polynomial Expansions for the Riemann Xi Function Dan Romik
    Orthogonal polynomial expansions for the Riemann xi function Dan Romik Author address: Department of Mathematics, University of California, Davis, One Shields Ave, Davis CA 95616, USA E-mail address: [email protected] Contents Chapter 1. Introduction 1 1.1. Background 1 1.2. Our new results: Tur´an'sprogram revisited and extended; expansion of Ξ(t) in new orthogonal polynomial bases 3 1.3. Previous work involving the polynomials fn 5 1.4. How to read this paper 6 1.5. Acknowledgements 6 Chapter 2. The Hermite expansion of Ξ(t) 7 2.1. The basic convergence result for the Hermite expansion 7 2.2. Preliminaries 8 2.3. Proof of Theorem 2.1 8 2.4. An asymptotic formula for the coefficients b2n 12 2.5. The Poisson flow, P´olya-De Bruijn flow and the De Bruijn-Newman constant 17 Chapter 3. Expansion of Ξ(t) in the polynomials fn 20 3.1. Main results 20 3.2. Proof of Theorem 3.1 21 3.3. Proof of Theorem 3.2 23 3.4. The Poisson flow associated with the fn-expansion 26 3.5. Evolution of the zeros under the Poisson flow 27 Chapter 4. Radial Fourier self-transforms 30 4.1. Radial Fourier self-transforms on Rd and their construction from balanced functions 30 4.2. The radial function A(r) associated to !(x) 31 4.3. An orthonormal basis for radial self-transforms 33 4.4. Constructing new balanced functions from old 35 4.5. The functions ν(x) and B(r) 35 4.6.
    [Show full text]
  • PHD THESIS a Class of Equivalent Problems Related to the Riemann
    PHDTHESIS Sadegh Nazardonyavi A Class of Equivalent Problems Related to the Riemann Hypothesis Tese submetida `aFaculdade de Ci^enciasda Universidade do Porto para obten¸c~ao do grau de Doutor em Matem´atica Departamento de Matem´atica Faculdade de Ci^enciasda Universidade do Porto 2013 To My Parents Acknowledgments I would like to thank my supervisor Prof. Semyon Yakubovich for all the guidance and support he provided for me during my studies at the University of Porto. I would also like to thank Professors: Bagher Nashvadian-Bakhsh, Abdolhamid Ri- azi, Abdolrasoul Pourabbas, Jos´eFerreira Alves for their teaching and supporting me in academic stuffs. Also I would like to thank Professors: Ana Paula Dias, Jos´eMiguel Urbano, Marc Baboulin, Jos´ePeter Gothen and Augusto Ferreira for the nice courses I had with them. I thank Professors J. C. Lagarias, C. Calderon, J. Stopple, and M. Wolf for useful discussions and sending us some relevant references. My sincere thanks to Profes- sor Jean-Louis Nicolas for careful reading some parts of the manuscript, helpful comments and suggestions which rather improved the presentation of the last chap- ter. Also I sincerely would like to thank Professor Paulo Eduardo Oliveira for his kindly assistances and advices as a coordinator of this PhD program. My thanks goes to all my friends which made a friendly environment, in particular Mohammad Soufi Neyestani for what he did before I came to Portugal until now. My gratitude goes to Funda¸c~aoCalouste Gulbenkian for the financial support dur- ing my PhD. I would like to thank all people who taught me something which are useful in my life but I have not mentioned their names.
    [Show full text]
  • Arxiv:2002.05234V1 [Cs.GR] 12 Feb 2020 Have Emerged
    VISUALIZING MODULAR FORMS DAVID LOWRY-DUDA Abstract. We describe practical methods to visualize modular forms. We survey several current visualizations. We then introduce an approach that can take advantage of colormaps in python's matplotlib library and describe an implementation. 1. Introduction 1.1. Motivation. Graphs of real-valued functions are ubiquitous and com- monly serve as a source of mathematical insight. But graphs of complex functions are simultaneously less common and more challenging to make. The reason is that the graph of a complex function is naturally a surface in four dimensions, and there are not many intuitive embeddings available to capture this surface within a 2d plot. In this article, we examine different methods for visualizing plots of mod- ular forms on congruent subgroups of SL(2; Z). These forms are highly symmetric functions and we should expect their plots to capture many dis- tinctive, highly symmetric features. In addition, we wish to take advantage of the broader capabilities that exist in the python/SageMath data visualization ecosystem. There are a vast number of color choices and colormaps implemented in terms of python's matplotlib library [Hun07]. While many of these are purely aesthetic, some offer color choices friendly to color blind viewers. Further, some are designed with knowledge of color theory and human cognition to be perceptually uniform with respect to human vision. We describe this further in x3. 1.2. Broad Overview of Complex Function Plotting. Over the last 20 years, different approaches towards representing graphs of complex functions arXiv:2002.05234v1 [cs.GR] 12 Feb 2020 have emerged.
    [Show full text]
  • An Analysis of the Lanczos Gamma Approximation
    AN ANALYSIS OF THE LANCZOS GAMMA APPROXIMATION by GLENDON RALPH PUGH B.Sc. (Mathematics) University of New Brunswick, 1992 M.Sc. (Mathematics) University of British Columbia, 1999 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard .................................. .................................. .................................. .................................. .................................. THE UNIVERSITY OF BRITISH COLUMBIA November 2004 c Glendon Ralph Pugh, 2004 In presenting this thesis in partial fulfillment of the requirements for an ad- vanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of Mathematics The University of British Columbia Vancouver, Canada Date Abstract This thesis is an analysis of C. Lanczos’ approximation of the classical gamma function Γ(z+1) as given in his 1964 paper A Precision Approximation of the Gamma Function [14]. The purposes of this study are: (i) to explain the details of Lanczos’ paper, including proofs of all claims made by the author; (ii) to address the question of how best to implement the approximation method in practice; and (iii) to generalize the methods used in the derivation of the approximation. At present there are a number of algorithms for approximating the gamma function.
    [Show full text]
  • Real-Time Visualization of Geometric Singularities Master’S Thesis by Stefan Kranich
    TECHNISCHE UNIVERSITÄT MÜNCHEN Department of Mathematics Real-time Visualization of Geometric Singularities Master’s thesis by Stefan Kranich Supervisor: Prof. Dr. Dr. J¨urgen Richter-Gebert Advisor: Prof. Dr. Dr. J¨urgen Richter-Gebert Submission date: September 17, 2012 Abstract Dynamic geometry is a branch of geometry concerned with movement of elements in geometric constructions. If we move an element of a construction and update the positions of the other elements accordingly, we traverse a continuum of instances of the construction, its configuration space. In many geometric constructions, interesting effects occur in the neighbourhood of geometric singularities, i.e. around points in configuration space at which ambiguous geometric operations of the construction branch. For example, at a geometric singularity two points may exchange their role of being the upper/lower point of intersection of two circles. On the other hand, we frequently see ambiguous elements jump unexpectedly from one branch to another in the majority of dynamic geometry software. If we want to explain such effects, we must consider complex coordinates for the elements of geometric constructions. Consequently, what often hinders us to fully comprehend the dynamics of geometric constructions is that our view is limited to the real perspective. In particular, the two-dimensional real plane exposed in dynamic geometry software is really just a section of the multidimensional complex configuration space which determines the dynamic behaviour of a construction. The goal of this thesis is to contribute to a better understanding of dynamic geom- etry by making the complex configuration space of certain constructions accessible to visual exploration in real-time.
    [Show full text]
  • The Riemann Hypothesis Frank Vega
    The Riemann Hypothesis Frank Vega To cite this version: Frank Vega. The Riemann Hypothesis. 2021. hal-02501243v11 HAL Id: hal-02501243 https://hal.archives-ouvertes.fr/hal-02501243v11 Preprint submitted on 17 May 2021 (v11), last revised 14 Jul 2021 (v14) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Riemann Hypothesis Frank Vega P 1 Abstract. Let's define δ(x) = ( q≤x q − log log x − B), where B ≈ 0:2614972128 is the Meissel-Mertens constant. The Robin theorem states that δ(x) changes sign infinitely often. Let's also define S(x) = θ(x) − x, where θ(x) is the Chebyshev function. A theorem due to Erhard Schmidt implies that S(x) changes sign infinitely often. Using the Nicolas theorem, we prove that when the inequalities δ(x) ≤ 0 and S(x) ≥ 0 are satisfied for some x ≥ 127, then the Riemann Hypothesis should be false. However, the Mertens second theorem states that limx!1 δ(x) = 0. Moreover, a result from the Gr¨onwall paper could be restated as limx!1 S(x) = 0. In this way, this work could mean a new step forward in the direction for finally solving the Riemann Hypothesis.
    [Show full text]