Generalized Riemann Hypothesis Léo Agélas
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Scalar Curvature and Geometrization Conjectures for 3-Manifolds
Comparison Geometry MSRI Publications Volume 30, 1997 Scalar Curvature and Geometrization Conjectures for 3-Manifolds MICHAEL T. ANDERSON Abstract. We first summarize very briefly the topology of 3-manifolds and the approach of Thurston towards their geometrization. After dis- cussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization Conjecture for closed oriented 3-manifolds. The final two sections present evidence for the validity of these conjectures and outline an approach toward their proof. Introduction In the late seventies and early eighties Thurston proved a number of very re- markable results on the existence of geometric structures on 3-manifolds. These results provide strong support for the profound conjecture, formulated by Thur- ston, that every compact 3-manifold admits a canonical decomposition into do- mains, each of which has a canonical geometric structure. For simplicity, we state the conjecture only for closed, oriented 3-manifolds. Geometrization Conjecture [Thurston 1982]. Let M be a closed , oriented, 2 prime 3-manifold. Then there is a finite collection of disjoint, embedded tori Ti 2 in M, such that each component of the complement M r Ti admits a geometric structure, i.e., a complete, locally homogeneous RiemannianS metric. A more detailed description of the conjecture and the terminology will be given in Section 1. A complete Riemannian manifold N is locally homogeneous if the universal cover N˜ is a complete homogenous manifold, that is, if the isometry group Isom N˜ acts transitively on N˜. It follows that N is isometric to N=˜ Γ, where Γ is a discrete subgroup of Isom N˜, which acts freely and properly discontinuously on N˜. -
Towards the Poincaré Conjecture and the Classification of 3-Manifolds John Milnor
Towards the Poincaré Conjecture and the Classification of 3-Manifolds John Milnor he Poincaré Conjecture was posed ninety- space SO(3)/I60 . Here SO(3) is the group of rota- nine years ago and may possibly have tions of Euclidean 3-space, and I60 is the subgroup been proved in the last few months. This consisting of those rotations which carry a regu- note will be an account of some of the lar icosahedron or dodecahedron onto itself (the Tmajor results over the past hundred unique simple group of order 60). This manifold years which have paved the way towards a proof has the homology of the 3-sphere, but its funda- and towards the even more ambitious project of mental group π1(SO(3)/I60) is a perfect group of classifying all compact 3-dimensional manifolds. order 120. He concluded the discussion by asking, The final paragraph provides a brief description of again translated into modern language: the latest developments, due to Grigory Perelman. A more serious discussion of Perelman’s work If a closed 3-dimensional manifold has will be provided in a subsequent note by Michael trivial fundamental group, must it be Anderson. homeomorphic to the 3-sphere? The conjecture that this is indeed the case has Poincaré’s Question come to be known as the Poincaré Conjecture. It At the very beginning of the twentieth century, has turned out to be an extraordinarily difficult Henri Poincaré (1854–1912) made an unwise claim, question, much harder than the corresponding which can be stated in modern language as follows: question in dimension five or more,2 and is a If a closed 3-dimensional manifold has key stumbling block in the effort to classify the homology of the sphere S3 , then it 3-dimensional manifolds. -
A Short and Simple Proof of the Riemann's Hypothesis
A Short and Simple Proof of the Riemann’s Hypothesis Charaf Ech-Chatbi To cite this version: Charaf Ech-Chatbi. A Short and Simple Proof of the Riemann’s Hypothesis. 2021. hal-03091429v10 HAL Id: hal-03091429 https://hal.archives-ouvertes.fr/hal-03091429v10 Preprint submitted on 5 Mar 2021 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. A Short and Simple Proof of the Riemann’s Hypothesis Charaf ECH-CHATBI ∗ Sunday 21 February 2021 Abstract We present a short and simple proof of the Riemann’s Hypothesis (RH) where only undergraduate mathematics is needed. Keywords: Riemann Hypothesis; Zeta function; Prime Numbers; Millennium Problems. MSC2020 Classification: 11Mxx, 11-XX, 26-XX, 30-xx. 1 The Riemann Hypothesis 1.1 The importance of the Riemann Hypothesis The prime number theorem gives us the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann’s 1859 paper[1], it asserts that all the ’non-trivial’ zeros of the zeta function are complex numbers with real part 1/2. 1.2 Riemann Zeta Function For a complex number s where ℜ(s) > 1, the Zeta function is defined as the sum of the following series: +∞ 1 ζ(s)= (1) ns n=1 X In his 1859 paper[1], Riemann went further and extended the zeta function ζ(s), by analytical continuation, to an absolutely convergent function in the half plane ℜ(s) > 0, minus a simple pole at s = 1: s +∞ {x} ζ(s)= − s dx (2) s − 1 xs+1 Z1 ∗One Raffles Quay, North Tower Level 35. -
Dirichlet's Theorem
Dirichlet's Theorem Austin Tran 2 June 2014 Abstract Shapiro's paper \On Primes in Arithmetic Progression" [11] gives a nontraditional proof for Dirichlet's Theorem, utilizing mostly elemen- tary algebraic number theory. Most other proofs of Dirichlet's theo- rem use Dirichlet characters and their respective L-functions, which fall under the field of analytic number theory. Dirichlet characters are functions defined over a group modulo n and are used to construct se- ries representing what are called Dirichlet L-functions in the complex plane. This paper provides the appropriate theoretical background in all number theory, and provides a detailed proof of Dirichlet's Theo- rem connecting analytic number theory from [11] and [9] and algebraic number theory from sources such as [1]. 1 Introduction Primes, and their distributions, have always been a central topic in num- ber theory. Sequences of primes, generated by functions and particularly polynomials, have captured the attention of famous mathematicians, unsur- prisingly including Euler and Dirichlet. In 1772 Euler famously noticed that the quadratic n2 −n+41 always produce primes for n from 1 to 41. However, it has been known for even longer that it is impossible for any polynomial to always evaluate to a prime. The proof is a straightforward one by contradic- tion, and we can give it here. Lemma 0: If f is a polynomial with integer coefficients, f(n) cannot be prime for all integers n. 1 Suppose such f exists. Then f(1) = p ≡ 0 (mod p). But f(1 + kp) is also equivalent to 0 modulo p for any k, so unless f(1 + kp) = p for all k, f does not produce all primes. -
An Explicit Formula for Dirichlet's L-Function
University of Tennessee at Chattanooga UTC Scholar Student Research, Creative Works, and Honors Theses Publications 5-2018 An explicit formula for Dirichlet's L-Function Shannon Michele Hyder University of Tennessee at Chattanooga, [email protected] Follow this and additional works at: https://scholar.utc.edu/honors-theses Part of the Mathematics Commons Recommended Citation Hyder, Shannon Michele, "An explicit formula for Dirichlet's L-Function" (2018). Honors Theses. This Theses is brought to you for free and open access by the Student Research, Creative Works, and Publications at UTC Scholar. It has been accepted for inclusion in Honors Theses by an authorized administrator of UTC Scholar. For more information, please contact [email protected]. An Explicit Formula for Dirichlet's L-Function Shannon M. Hyder Departmental Honors Thesis The University of Tennessee at Chattanooga Department of Mathematics Thesis Director: Dr. Andrew Ledoan Examination Date: April 9, 2018 Members of Examination Committee Dr. Andrew Ledoan Dr. Cuilan Gao Dr. Roger Nichols c 2018 Shannon M. Hyder ALL RIGHTS RESERVED i Abstract An Explicit Formula for Dirichlet's L-Function by Shannon M. Hyder The Riemann zeta function has a deep connection to the distribution of primes. In 1911 Landau proved that, for every fixed x > 1, X T xρ = − Λ(x) + O(log T ) 2π 0<γ≤T as T ! 1. Here ρ = β + iγ denotes a complex zero of the zeta function and Λ(x) is an extension of the usual von Mangoldt function, so that Λ(x) = log p if x is a positive integral power of a prime p and Λ(x) = 0 for all other real values of x. -
Aspects of Quantum Field Theory and Number Theory, Quantum Field Theory and Riemann Zeros, © July 02, 2015
juan guillermo dueñas luna ASPECTSOFQUANTUMFIELDTHEORYAND NUMBERTHEORY ASPECTSOFQUANTUMFIELDTHEORYANDNUMBER THEORY juan guillermo dueñas luna Quantum Field Theory and Riemann Zeros Theoretical Physics Department. Brazilian Center For Physics Research. July 02, 2015. Juan Guillermo Dueñas Luna: Aspects of Quantum Field Theory and Number Theory, Quantum Field Theory and Riemann Zeros, © July 02, 2015. supervisors: Nami Fux Svaiter. location: Rio de Janeiro. time frame: July 02, 2015. ABSTRACT The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line Re(s) = 1=2. Motivated by the Hilbert- Pólya conjecture which states that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in the light of spectral theory, a lot of activity has been devoted to establish a bridge between number theory and physics. Using the techniques of the spectral zeta function we show that prime numbers and the Rie- mann zeros have a different behaviour as the spectrum of a linear operator associated to a system with countable infinite number of degrees of freedom. To explore more connections between quantum field theory and number theory we studied three systems involving these two sequences of numbers. First, we discuss the renormalized zero-point energy for a massive scalar field such that the Riemann zeros appear in the spectrum of the vacuum modes. This scalar field is defined in a (d + 1)-dimensional flat space-time, assuming that one of the coordinates lies in an finite interval [0, a]. For even dimensional space-time we found a finite reg- ularized energy density, while for odd dimensional space-time we are forced to introduce mass counterterms to define a renormalized vacuum energy. -
The Process of Student Cognition in Constructing Mathematical Conjecture
ISSN 2087-8885 E-ISSN 2407-0610 Journal on Mathematics Education Volume 9, No. 1, January 2018, pp. 15-26 THE PROCESS OF STUDENT COGNITION IN CONSTRUCTING MATHEMATICAL CONJECTURE I Wayan Puja Astawa1, I Ketut Budayasa2, Dwi Juniati2 1Ganesha University of Education, Jalan Udayana 11, Singaraja, Indonesia 2State University of Surabaya, Ketintang, Surabaya, Indonesia Email: [email protected] Abstract This research aims to describe the process of student cognition in constructing mathematical conjecture. Many researchers have studied this process but without giving a detailed explanation of how students understand the information to construct a mathematical conjecture. The researchers focus their analysis on how to construct and prove the conjecture. This article discusses the process of student cognition in constructing mathematical conjecture from the very beginning of the process. The process is studied through qualitative research involving six students from the Mathematics Education Department in the Ganesha University of Education. The process of student cognition in constructing mathematical conjecture is grouped into five different stages. The stages consist of understanding the problem, exploring the problem, formulating conjecture, justifying conjecture, and proving conjecture. In addition, details of the process of the students’ cognition in each stage are also discussed. Keywords: Process of Student Cognition, Mathematical Conjecture, qualitative research Abstrak Penelitian ini bertujuan untuk mendeskripsikan proses kognisi mahasiswa dalam mengonstruksi konjektur matematika. Beberapa peneliti mengkaji proses-proses ini tanpa menjelaskan bagaimana siswa memahami informasi untuk mengonstruksi konjektur. Para peneliti dalam analisisnya menekankan bagaimana mengonstruksi dan membuktikan konjektur. Artikel ini membahas proses kognisi mahasiswa dalam mengonstruksi konjektur matematika dari tahap paling awal. -
RIEMANN's HYPOTHESIS 1. Gauss There Are 4 Prime Numbers Less
RIEMANN'S HYPOTHESIS BRIAN CONREY 1. Gauss There are 4 prime numbers less than 10; there are 25 primes less than 100; there are 168 primes less than 1000, and 1229 primes less than 10000. At what rate do the primes thin out? Today we use the notation π(x) to denote the number of primes less than or equal to x; so π(1000) = 168. Carl Friedrich Gauss in an 1849 letter to his former student Encke provided us with the answer to this question. Gauss described his work from 58 years earlier (when he was 15 or 16) where he came to the conclusion that the likelihood of a number n being prime, without knowing anything about it except its size, is 1 : log n Since log 10 = 2:303 ::: the means that about 1 in 16 seven digit numbers are prime and the 100 digit primes are spaced apart by about 230. Gauss came to his conclusion empirically: he kept statistics on how many primes there are in each sequence of 100 numbers all the way up to 3 million or so! He claimed that he could count the primes in a chiliad (a block of 1000) in 15 minutes! Thus we expect that 1 1 1 1 π(N) ≈ + + + ··· + : log 2 log 3 log 4 log N This is within a constant of Z N du li(N) = 0 log u so Gauss' conjecture may be expressed as π(N) = li(N) + E(N) Date: January 27, 2015. 1 2 BRIAN CONREY where E(N) is an error term. -
History of Mathematics
History of Mathematics James Tattersall, Providence College (Chair) Janet Beery, University of Redlands Robert E. Bradley, Adelphi University V. Frederick Rickey, United States Military Academy Lawrence Shirley, Towson University Introduction. There are many excellent reasons to study the history of mathematics. It helps students develop a deeper understanding of the mathematics they have already studied by seeing how it was developed over time and in various places. It encourages creative and flexible thinking by allowing students to see historical evidence that there are different and perfectly valid ways to view concepts and to carry out computations. Ideally, a History of Mathematics course should be a part of every mathematics major program. A course taught at the sophomore-level allows mathematics students to see the great wealth of mathematics that lies before them and encourages them to continue studying the subject. A one- or two-semester course taught at the senior level can dig deeper into the history of mathematics, incorporating many ideas from the 19th and 20th centuries that could only be approached with difficulty by less prepared students. Such a senior-level course might be a capstone experience taught in a seminar format. It would be wonderful for students, especially those planning to become middle school or high school mathematics teachers, to have the opportunity to take advantage of both options. We also encourage History of Mathematics courses taught to entering students interested in mathematics, perhaps as First Year or Honors Seminars; to general education students at any level; and to junior and senior mathematics majors and minors. Ideally, mathematics history would be incorporated seamlessly into all courses in the undergraduate mathematics curriculum in addition to being addressed in a few courses of the type we have listed. -
Arxiv:2102.04549V2 [Math.GT] 26 Jun 2021 Ieade Nih Notesrcueo -Aiod.Alteeconj Polyto These and Norm All Thurston to the 3-Manifolds
SURVEY ON L2-INVARIANTS AND 3-MANIFOLDS LUCK,¨ W. Abstract. In this paper give a survey about L2-invariants focusing on 3- manifolds. 0. Introduction The theory of L2-invariants was triggered by Atiyah in his paper on the L2-index theorem [4]. He followed the general principle to consider a classical invariant of a compact manifold and to define its analog for the universal covering taking the action of the fundamental group into account. Its application to (co)homology, Betti numbers and Reidemeister torsion led to the notions of L2-(cohomology), L2-Betti numbers and L2-torsion. Since then L2-invariants were developed much further. They already had and will continue to have striking, surprizing, and deep impact on questions and problems of fields, for some of which one would not expect any relation, e.g., to algebra, differential geometry, global analysis, group theory, topology, transformation groups, and von Neumann algebras. The theory of 3-manifolds has also a quite impressive history culminating in the work of Waldhausen and in particular of Thurston and much later in the proof of the Geometrization Conjecture by Perelman and of the Virtual Fibration Conjecture by Agol. It is amazing how many beautiful, easy to comprehend, and deep theorems, which often represent the best result one can hope for, have been proved for 3- manifolds. The motivating question of this survey paper is: What happens if these two prominent and successful areas meet one another? The answer will be: Something very interesting. In Section 1 we give a brief overview over basics about 3-manifolds, which of course can be skipped by someone who has already some background. -
Prime Numbers and the Riemann Hypothesis
Prime numbers and the Riemann hypothesis Tatenda Kubalalika October 2, 2019 ABSTRACT. Denote by ζ the Riemann zeta function. By considering the related prime zeta function, we demonstrate in this note that ζ(s) 6= 0 for <(s) > 1=2, which proves the Riemann hypothesis. Keywords and phrases: Prime zeta function, Riemann zeta function, Riemann hypothesis, proof. 2010 Mathematics Subject Classifications: 11M26, 11M06. Introduction. Prime numbers have fascinated mathematicians since the ancient Greeks, and Euclid provided the first proof of their infinitude. Central to this subject is some innocent-looking infinite series known as the Riemann zeta function. This is is a function of the complex variable s, defined in the half-plane <(s) > 1 by 1 X ζ(s) := n−s n=1 and in the whole complex plane by analytic continuation. Euler noticed that the above series can Q −s −1 be expressed as a product p(1 − p ) over the entire set of primes, which entails that ζ(s) 6= 0 for <(s) > 1. As shown by Riemann [2], ζ(s) extends to C as a meromorphic function with only a simple pole at s = 1, with residue 1, and satisfies the functional equation ξ(s) = ξ(1 − s), 1 −z=2 1 R 1 −x w−1 where ξ(z) = 2 z(z − 1)π Γ( 2 z)ζ(z) and Γ(w) = 0 e x dx. From the functional equation and the relationship between Γ and the sine function, it can be easily noticed that 8n 2 N one has ζ(−2n) = 0, hence the negative even integers are referred to as the trivial zeros of ζ in the literature. -
“Character” in Dirichlet's Theorem on Primes in an Arithmetic Progression
The concept of “character” in Dirichlet’s theorem on primes in an arithmetic progression∗ Jeremy Avigad and Rebecca Morris October 30, 2018 Abstract In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses of Dirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye towards understanding some of the pragmatic pres- sures that shaped the evolution of modern mathematical method. Contents 1 Introduction 2 2 Functions in the nineteenth century 3 2.1 Thegeneralizationofthefunctionconcept . 3 2.2 Theevolutionofthefunctionconcept . 10 3 Dirichlet’s theorem 12 3.1 Euler’s proof that there are infinitely many primes . 13 3.2 Groupcharacters ........................... 14 3.3 Dirichlet characters and L-series .................. 17 3.4 ContemporaryproofsofDirichlet’stheorem . 18 arXiv:1209.3657v3 [math.HO] 26 Jun 2013 4 Modern aspects of contemporary proofs 19 5 Dirichlet’s original proof 22 ∗This essay draws extensively on the second author’s Carnegie Mellon MS thesis [64]. We are grateful to Michael Detlefsen and the participants in his Ideals of Proof workshop, which provided feedback on portions of this material in July, 2011, and to Jeremy Gray for helpful comments. Avigad’s work has been partially supported by National Science Foundation grant DMS-1068829 and Air Force Office of Scientific Research grant FA9550-12-1-0370. 1 6 Later presentations 28 6.1 Dirichlet................................ 28 6.2 Dedekind ............................... 30 6.3 Weber ................................. 34 6.4 DelaVall´ee-Poussin ......................... 36 6.5 Hadamard..............................