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[Math.NT] 10 May 2005
Journal de Th´eorie des Nombres de Bordeaux 00 (XXXX), 000–000 Restriction theory of the Selberg sieve, with applications par Ben GREEN et Terence TAO Resum´ e.´ Le crible de Selberg fournit des majorants pour cer- taines suites arithm´etiques, comme les nombres premiers et les nombres premiers jumeaux. Nous demontrons un th´eor`eme de restriction L2-Lp pour les majorants de ce type. Comme ap- plication imm´ediate, nous consid´erons l’estimation des sommes exponentielles sur les k-uplets premiers. Soient les entiers posi- tifs a1,...,ak et b1,...,bk. Ecrit h(θ) := n∈X e(nθ), ou X est l’ensemble de tous n 6 N tel que tousP les nombres a1n + b1,...,akn+bk sont premiers. Nous obtenons les bornes sup´erieures pour h p T , p > 2, qui sont (en supposant la v´erit´ede la con- k kL ( ) jecture de Hardy et Littlewood sur les k-uplets premiers) d’ordre de magnitude correct. Une autre application est la suivante. En utilisant les th´eor`emes de Chen et de Roth et un «principe de transf´erence », nous demontrons qu’il existe une infinit´ede suites arithm´etiques p1 < p2 < p3 de nombres premiers, telles que cha- cun pi + 2 est premier ou un produit de deux nombres premier. Abstract. The Selberg sieve provides majorants for certain arith- metic sequences, such as the primes and the twin primes. We prove an L2–Lp restriction theorem for majorants of this type. An im- mediate application is to the estimation of exponential sums over prime k-tuples. -
On Sufficient Conditions for the Existence of Twin Values in Sieves
On Sufficient Conditions for the Existence of Twin Values in Sieves over the Natural Numbers by Luke Szramowski Submitted in Partial Fulfillment of the Requirements For the Degree of Masters of Science in the Mathematics Program Youngstown State University May, 2020 On Sufficient Conditions for the Existence of Twin Values in Sieves over the Natural Numbers Luke Szramowski I hereby release this thesis to the public. I understand that this thesis will be made available from the OhioLINK ETD Center and the Maag Library Circulation Desk for public access. I also authorize the University or other individuals to make copies of this thesis as needed for scholarly research. Signature: Luke Szramowski, Student Date Approvals: Dr. Eric Wingler, Thesis Advisor Date Dr. Thomas Wakefield, Committee Member Date Dr. Thomas Madsen, Committee Member Date Dr. Salvador A. Sanders, Dean of Graduate Studies Date Abstract For many years, a major question in sieve theory has been determining whether or not a sieve produces infinitely many values which are exactly two apart. In this paper, we will discuss a new result in sieve theory, which will give sufficient conditions for the existence of values which are exactly two apart. We will also show a direct application of this theorem on an existing sieve as well as detailing attempts to apply the theorem to the Sieve of Eratosthenes. iii Contents 1 Introduction 1 2 Preliminary Material 1 3 Sieves 5 3.1 The Sieve of Eratosthenes . 5 3.2 The Block Sieve . 9 3.3 The Finite Block Sieve . 12 3.4 The Sieve of Joseph Flavius . -
Sieve Theory and Small Gaps Between Primes: Introduction
Sieve theory and small gaps between primes: Introduction Andrew V. Sutherland MASSACHUSETTS INSTITUTE OF TECHNOLOGY (on behalf of D.H.J. Polymath) Explicit Methods in Number Theory MATHEMATISCHES FORSCHUNGSINSTITUT OBERWOLFACH July 6, 2015 A quick historical overview pn+m − pn ∆m := lim inf Hm := lim inf (pn+m − pn) n!1 log pn n!1 Twin Prime Conjecture: H1 = 2 Prime Tuples Conjecture: Hm ∼ m log m 1896 Hadamard–Vallee´ Poussin ∆1 ≤ 1 1926 Hardy–Littlewood ∆1 ≤ 2=3 under GRH 1940 Rankin ∆1 ≤ 3=5 under GRH 1940 Erdos˝ ∆1 < 1 1956 Ricci ∆1 ≤ 15=16 1965 Bombieri–Davenport ∆1 ≤ 1=2, ∆m ≤ m − 1=2 ········· 1988 Maier ∆1 < 0:2485. 2005 Goldston-Pintz-Yıldırım ∆1 = 0, ∆m ≤ m − 1, EH ) H1 ≤ 16 2013 Zhang H1 < 70; 000; 000 2013 Polymath 8a H1 ≤ 4680 3 4m 2013 Maynard-Tao H1 ≤ 600, Hm m e , EH ) H1 ≤ 12 3:815m 2014 Polymath 8b H1 ≤ 246, Hm e , GEH ) H1 ≤ 6 H2 ≤ 398; 130, H3 ≤ 24; 797; 814,... The prime number theorem in arithmetic progressions Define the weighted prime counting functions1 X X Θ(x) := log p; Θ(x; q; a) := log p: prime p ≤ x prime p ≤ x p ≡ a mod q Then Θ(x) ∼ x (the prime number theorem), and for a ? q, x Θ(x; q; a) ∼ : φ(q) We are interested in the discrepancy between these two quantities. −x x x Clearly φ(q) ≤ Θ(x; q; a) − φ(q) ≤ q + 1 log x, and for any Q < x, X x X 2x log x x max Θ(x; q; a) − ≤ + x(log x)2: a?q φ(q) q φ(q) q ≤ Q q≤Q 1 P One can also use (x) := n≤x Λ(n), where Λ(n) is the von Mangoldt function. -
Explicit Counting of Ideals and a Brun-Titchmarsh Inequality for The
EXPLICIT COUNTING OF IDEALS AND A BRUN-TITCHMARSH INEQUALITY FOR THE CHEBOTAREV DENSITY THEOREM KORNEEL DEBAENE Abstract. We prove a bound on the number of primes with a given split- ting behaviour in a given field extension. This bound generalises the Brun- Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg’s Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a conse- quence of this result, we deduce an explicit estimate for the number of ideals of norm up to x. 1. Introduction and statement of results Let q and a be coprime integers, and let π(x,q,a) be the number of primes not exceeding x which are congruent to a mod q. It is well known that for fixed q, π(x,q,a) 1 lim = . x→∞ π(x) φ(q) A major drawback of this result is the lack of an effective error term, and hence the impossibility to let q tend to infinity with x. The Brun-Titchmarsh inequality remedies the situation, if one is willing to accept a less precise relation in return for an effective range of q. It states that 2 x π(x,q,a) , for any x q. ≤ φ(q) log x/q ≥ While originally proven with 2 + ε in place of 2, the above formulation was proven by Montgomery and Vaughan [16] in 1973. Further improvement on this constant seems out of reach, indeed, if one could prove that the inequality holds with 2 δ arXiv:1611.10103v2 [math.NT] 9 Mar 2017 in place of 2, for any positive δ, one could deduce from this that there are no Siegel− 1 zeros. -
An Introduction to Sieve Methods and Their Applications Alina Carmen Cojocaru , M
Cambridge University Press 978-0-521-84816-9 — An Introduction to Sieve Methods and Their Applications Alina Carmen Cojocaru , M. Ram Murty Frontmatter More Information An Introduction to Sieve Methods and Their Applications © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-84816-9 — An Introduction to Sieve Methods and Their Applications Alina Carmen Cojocaru , M. Ram Murty Frontmatter More Information LONDON MATHEMATICAL SOCIETY STUDENT TEXTS Managing editor: Professor J. W. Bruce, Department of Mathematics, University of Hull, UK 3 Local fields, J. W. S. CASSELS 4 An introduction to twistor theory: Second edition, S. A. HUGGETT & K. P. TOD 5 Introduction to general relativity, L. P. HUGHSTON & K. P. TOD 8 Summing and nuclear norms in Banach space theory, G. J. O. JAMESON 9 Automorphisms of surfaces after Nielsen and Thurston, A. CASSON & S. BLEILER 11 Spacetime and singularities, G. NABER 12 Undergraduate algebraic geometry, MILES REID 13 An introduction to Hankel operators, J. R. PARTINGTON 15 Presentations of groups: Second edition, D. L. JOHNSON 17 Aspects of quantum field theory in curved spacetime, S. A. FULLING 18 Braids and coverings: selected topics, VAGN LUNDSGAARD HANSEN 20 Communication theory, C. M. GOLDIE & R. G. E. PINCH 21 Representations of finite groups of Lie type, FRANCOIS DIGNE & JEAN MICHEL 22 Designs, graphs, codes, and their links, P. J. CAMERON & J. H. VAN LINT 23 Complex algebraic curves, FRANCES KIRWAN 24 Lectures on elliptic curves, J. W. S CASSELS 26 An introduction to the theory of L-functions and Eisenstein series, H. HIDA 27 Hilbert Space: compact operators and the trace theorem, J. -
Asymptotic Sieve for Primes 1043
Annals of Mathematics, 148 (1998), 1041–1065 Asymptotic sieve for primes By John Friedlander and Henryk Iwaniec 1. Heuristics and statement of results For a long time it was believed that sieve methods might be incapable of achieving the goal for which they had been created, the detection of prime numbers. Indeed, it was shown [S], [B] to be inevitable that, in the sieve’s original framework, no such result was possible although one could come tan- talizingly close. This general limitation is recognized as the “parity problem” of sieve theory. More recently, beginning with the work [IJ], this goal has in certain cases become possible by adapting the sieve machinery to enable it to take advantage of the input of additional analytic data. There have been a number of recent developments in this regard, for example [H], [DFI], [FI], and several recent works of R.C. Baker and G. Harman. The story however is far from finished. In this paper we consider a sequence of real nonnegative numbers (1.1) = (a ) A n with the purpose of showing an asymptotic formula for (1.2) S(x)= anΛ(n) nX6x where Λ(n) denotes the von Mangoldt function. The sequence can be quite A thin although not arbitrarily so. Setting arXiv:math/9811186v1 [math.NT] 1 Nov 1998 (1.3) A(x)= an nX6x we require that A(x) be slightly larger than A(√x); precisely (1.4) A(x) A(√x)(log x)2 . ≫ *JF was supported in part by NSERC grant A5123 and HI was supported in part by NSF grant DMS-9500797. -
A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture
A FRIENDLY INTRO TO SIEVES WITH A LOOK TOWARDS RECENT PROGRESS ON THE TWIN PRIMES CONJECTURE DAVID LOWRY-DUDA This is an extension and background to a talk I gave on 9 October 2013 to the Brown Graduate Student Seminar, called `A friendly intro to sieves with a look towards recent progress on the twin primes conjecture.' During the talk, I mention several sieves, some with a lot of detail and some with very little detail. I also discuss several results and built upon many sources. I'll provide missing details and/or sources for additional reading here. Furthermore, I like this talk, so I think it's worth preserving. 1. Introduction We talk about sieves and primes. Long, long ago, Euclid famously proved the in- finitude of primes (≈ 300 B.C.). Although he didn't show it, the stronger statement that the sum of the reciprocals of the primes diverges is true: X 1 ! 1; p p where the sum is over primes. Proof. Suppose that the sum converged. Then there is some k such that 1 X 1 1 < : pi 2 i=k+1 Qk Suppose that Q := i=1 pi is the product of the primes up to pk. Then the integers 1 + Qn are relatively prime to the primes in Q, and so are only made up of the primes pk+1;:::. This means that 1 !t X 1 X X 1 ≤ < 2; 1 + Qn pi n=1 t≥0 i>k where the first inequality is true since all the terms on the left appear in the middle (think prime factorizations and the distributive law), and the second inequality is true because it's bounded by the geometric series with ratio 1=2. -
Which Polynomials Represent Infinitely Many Primes?
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 14, Number 1 (2018), pp. 161-180 © Research India Publications http://www.ripublication.com/gjpam.htm Which polynomials represent infinitely many primes? Feng Sui Liu Department of Mathematics, NanChang University, NanChang, China. Abstract In this paper a new arithmetical model has been found by extending both basic operations + and × into finite sets of natural numbers. In this model we invent a recursive algorithm on sets of natural numbers to reformulate Eratosthene’s sieve. By this algorithm we obtain a recursive sequence of sets, which converges to the set of all natural numbers x such that x2 + 1 is prime. The corresponding cardinal sequence is strictly increasing. Test that the cardinal function is continuous with respect to an order topology, we immediately prove that there are infinitely many natural numbers x such that x2 + 1 is prime. Based on the reasoning paradigm above we further prove a general result: if an integer valued polynomial of degree k represents at least k +1 positive primes, then this polynomial represents infinitely many primes. AMS subject classification: Primary 11N32; Secondary 11N35, 11U09,11Y16, 11B37. Keywords: primes in polynomial, twin primes, arithmetical model, recursive sieve method, limit of sequence of sets, Ross-Littwood paradox, parity problem. 1. Introduction Whether an integer valued polynomial represents infinitely many primes or not, this is an intriguing question. Except some obvious exceptions one conjectured that the answer is yes. Example 1.1. Bouniakowsky [2], Schinzel [35], Bateman and Horn [1]. In this paper a recursive algorithm or sieve method adds some exotic structures to the sets of natural numbers, which allow us to answer the question: which polynomials represent infinitely many primes? 2 Feng Sui Liu In 1837, G.L. -
There Are Infinitely Many Mersenne Primes
Scan to know paper details and author's profile There are Infinitely Many Mersenne Primes Fengsui Liu ZheJiang University Alumnus ABSTRACT From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained. Then we invent a recursive sieve method for exponents of Mersenne primes. This is a novel algorithm on sets of natural numbers. The algorithm mechanically yields a sequence of sets of exponents of almost Mersenne primes, which converge to the set of exponents of all Mersenne primes. The corresponding cardinal sequence is strictly increasing. We capture a particular order topological structure of the set of exponents of all Mersenne primes. The existing theory of this structure allows us to prove that the set of exponents of all Mersenne primes is an infnite set . Keywords and phrases: Mersenne prime conjecture, modulo algorithm, recursive sieve method, limit of sequence of sets. Classification: FOR Code: 010299 Language: English LJP Copyright ID: 925622 Print ISSN: 2631-8490 Online ISSN: 2631-8504 London Journal of Research in Science: Natural and Formal 465U Volume 20 | Issue 3 | Compilation 1.0 © 2020. Fengsui Liu. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncom-mercial 4.0 Unported License http://creativecommons.org/licenses/by-nc/4.0/), permitting all noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited. There are Infinitely Many Mersenne Primes Fengsui Liu ____________________________________________ ABSTRACT From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained. -
Eureka Issue 61
Eureka 61 A Journal of The Archimedeans Cambridge University Mathematical Society Editors: Philipp Legner and Anja Komatar © The Archimedeans (see page 94 for details) Do not copy or reprint any parts without permission. October 2011 Editorial Eureka Reinvented… efore reading any part of this issue of Eureka, you will have noticed The Team two big changes we have made: Eureka is now published in full col- our, and printed on a larger paper size than usual. We felt that, with Philipp Legner Design and Bthe internet being an increasingly large resource for mathematical articles of Illustrations all kinds, it was necessary to offer something new and exciting to keep Eu- reka as successful as it has been in the past. We moved away from the classic Anja Komatar Submissions LATEX-look, which is so common in the scientific community, to a modern, more engaging, and more entertaining design, while being conscious not to Sean Moss lose any of the mathematical clarity and rigour. Corporate Ben Millwood To make full use of the new design possibilities, many of this issue’s articles Publicity are based around mathematical images: from fractal modelling in financial Lu Zou markets (page 14) to computer rendered pictures (page 38) and mathemati- Subscriptions cal origami (page 20). The Showroom (page 46) uncovers the fundamental role pictures have in mathematics, including patterns, graphs, functions and fractals. This issue includes a wide variety of mathematical articles, problems and puzzles, diagrams, movie and book reviews. Some are more entertaining, such as Bayesian Bets (page 10), some are more technical, such as Impossible Integrals (page 80), or more philosophical, such as How to teach Physics to Mathematicians (page 42). -
Every Sufficiently Large Even Number Is the Sum of Two Primes
EVERY SUFFICIENTLY LARGE EVEN NUMBER IS THE SUM OF TWO PRIMES Ricardo G. Barca Abstract The binary Goldbach conjecture asserts that every even integer greater than 4 is the sum of two primes. In this 2 paper, we prove that there exists an integer Kα such that every even integer x>pk can be expressed as the sum of two primes, where pk is the kth prime number and k > Kα. To prove this statement, we begin by introducing a type of double sieve of Eratosthenes as follows. Given a positive even integer x> 4, we sift from [1,x] all those elements that are congruents to 0 modulo p or congruents to x modulo p, where p is a prime less than √x. Therefore, any integer in the interval [√x,x] that remains unsifted is a prime q for which either x q = 1 or x q is also a prime. − − Then, we introduce a new way of formulating a sieve, which we call the sequence of k-tuples of remainders. By means of this tool, we prove that there exists an integer Kα > 5 such that pk/2 is a lower bound for the sifting 2 2 function of this sieve, for every even number x that satisfies pk <x<pk+1, where k > Kα, which implies that 2 x>pk (k > Kα) can be expressed as the sum of two primes. 1 Introduction 1.1 The sieve method and the Goldbach problem In 1742, Goldbach wrote a letter to his friend Euler telling about a conjecture involving prime numbers. -
January 2002 Prizes and Awards
January 2002 Prizes and Awards 4:25 p.m., Monday, January 7, 2002 PROGRAM OPENING REMARKS Ann E. Watkins, President Mathematical Association of America BECKENBACH BOOK PRIZE Mathematical Association of America BÔCHER MEMORIAL PRIZE American Mathematical Society LEVI L. CONANT PRIZE American Mathematical Society LOUISE HAY AWARD FOR CONTRIBUTIONS TO MATHEMATICS EDUCATION Association for Women in Mathematics ALICE T. S CHAFER PRIZE FOR EXCELLENCE IN MATHEMATICS BY AN UNDERGRADUATE WOMAN Association for Women in Mathematics CHAUVENET PRIZE Mathematical Association of America FRANK NELSON COLE PRIZE IN NUMBER THEORY American Mathematical Society AWARD FOR DISTINGUISHED PUBLIC SERVICE American Mathematical Society CERTIFICATES OF MERITORIOUS SERVICE Mathematical Association of America LEROY P. S TEELE PRIZE FOR MATHEMATICAL EXPOSITION American Mathematical Society LEROY P. S TEELE PRIZE FOR SEMINAL CONTRIBUTION TO RESEARCH American Mathematical Society LEROY P. S TEELE PRIZE FOR LIFETIME ACHIEVEMENT American Mathematical Society DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY TEACHING OF MATHEMATICS Mathematical Association of America CLOSING REMARKS Hyman Bass, President American Mathematical Society MATHEMATICAL ASSOCIATION OF AMERICA BECKENBACH BOOK PRIZE The Beckenbach Book Prize, established in 1986, is the successor to the MAA Book Prize. It is named for the late Edwin Beckenbach, a long-time leader in the publica- tions program of the Association and a well-known professor of mathematics at the University of California at Los Angeles. The prize is awarded for distinguished, innov- ative books published by the Association. Citation Joseph Kirtland Identification Numbers and Check Digit Schemes MAA Classroom Resource Materials Series This book exploits a ubiquitous feature of daily life, identification numbers, to develop a variety of mathematical ideas, such as modular arithmetic, functions, permutations, groups, and symmetries.