Efficient Prime Counting and the Chebyshev Primes
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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∗. -
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. -
THE DISTRIBUTION of PRIME NUMBERS Andrew Granville and K
THE DISTRIBUTION OF PRIME NUMBERS Andrew Granville and K. Soundararajan 0. Preface: Seduced by zeros 0.1. The prime number theorem, a brief history. A prime number is a positive integer which has no positive integer factors other than 1 and itself. It is difficult to determine directly from this definition whether there are many primes, indeed whether there are infinitely many. Euclid described in his Elements, an ancient Greek proof that there are infinitely many primes, a proof by contradiction, that today highlights for us the depth of abstract thinking in antiquity. So the next question is to quantify how many primes there are up to a given point. By studying tables of primes, Gauss understood, as a boy of 15 or 16 (in 1792 or 1 1793), that the primes occur with density log x at around x. In other words Z X 1 x dt π(x) := #{primes ≤ x} is approximately ≈ Li(x) where Li(x) := . log n log t n≤x 2 This leads to the conjecture that π(x)/Li(x) → 1 as x → ∞, which is hard to interpret x since Li(x) is not such a natural function. One can show that Li(x)/ log x → 1 as x → ∞, x so we can rephrase our conjecture as π(x)/ log x → 1 as x → ∞, or in less cumbersome notation that x (0.1.1) π(x) ∼ . log x It is not easy to find a strategy to prove such a result. Since primes are those integers with no prime factors less than√ or equal to their square-root, one obvious approach to counting the number of primes√ in ( x, x] is to try to estimate the number of integers up to x, with no prime factors ≤ x. -
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. -
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..................................... -
Survey Report
YouGov / The Sun Survey Results Sample Size: 1923 GB Adults Fieldwork: 31st August - 1st September 2010 Voting intention Gender Age Social grade Region Lib Rest of Midlands Total Con Lab Male Female 18-24 25-39 40-59 60+ ABC1 C2DE London North Scotland Dem South / Wales Weighted Sample 1923 635 557 179 938 985 232 491 661 539 1079 814 247 625 414 470 167 Unweighted Sample 1923 622 545 194 916 1007 124 505 779 515 1244 649 239 661 370 422 231 % %%% % % %%%%% % % % % % % 10 - 11 31 Aug - May 1 Sept 2007 2010 Former Prime Minister Tony Blair is due to publish his memoirs Thinking back to his time in office, do you think Tony Blair...? Was a very good Prime Minister and achieved many 11 10 4 22 7 11 8 5 11 11 8 9 11 12 8 11 9 10 successes Was a fairly good Prime Minister - his successes 38 37 23 58 42 40 35 38 43 39 29 38 35 35 32 41 43 32 outnumbered his failures Was a fairlyyp poor Prime Minister - his failures outnumbered 28 21 28 11 27 22 20 21 21 19 22 23 19 17 25 18 18 24 his successes Was a very poor Prime Minister and can be blamed for many 18 25 42 7 22 23 27 14 15 26 39 25 26 28 27 23 24 26 failures Don't know 5 7 3 2 3 4 10 21 10 4 2 5 10 9 8 6 6 8 Since the second world war, six Prime Ministers have served for five years or longer. -
Sparse Interpolation Over Finite Fields Via Low-Order Roots of Unity
Sparse interpolation over finite fields via low-order roots of unity Andrew Arnold Mark Giesbrecht Cheriton School of Computer Science Cheriton School of Computer Science University of Waterloo University of Waterloo [email protected] [email protected] Daniel S. Roche Computer Science Department United States Naval Academy [email protected] August 21, 2018 Abstract We present a new Monte Carlo algorithm for the interpolation of a straight-line program as a sparse polynomial f over an arbitrary finite field of size q. We assume a priori bounds D and T are given on the degree and number of terms of f. The approach presented in this paper is a hybrid of the diversified and recursive interpolation algorithms, the two previous fastest known probabilistic methods for this problem. By making effective use of the information contained in the coefficients themselves, this new algorithm improves on the bit complexity of previous methods by a “soft-Oh” factor of T , log D, or log q. 1 Introduction arXiv:1401.4744v2 [cs.SC] 2 May 2014 Let Fq be a finite field of size q and consider a “sparse” polynomial t ei F f = i=1 ciz ∈ q[z], (1.1) P where the c1,...,ct ∈ Fq are nonzero and the exponents e1,...,et ∈ Z≥0 are distinct. Suppose f is provided as a straight-line program, a simple branch-free program which evaluates f at any point (see below for a formal definition). Suppose also that we are given an upper bound D on the degree of f and an upper bound T on the number of non-zero terms t of f. -
Numbers 1 to 100
Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number) -
Heuristics on Class Groups: Some Good Primes Are Not Too Good
MATHEMATICSof computation VOLUME63, NUMBER207 JULY 1994, PAGES 329-334 HEURISTICS ON CLASS GROUPS: SOME GOOD PRIMES ARE NOT TOO GOOD HENRI COHEN AND JACQUESMARTINET Abstract. We correct some too optimistic predictions in an earlier paper of ours. 1 When generalizing to arbitrary relative extensions of number fields the Cohen and Lenstra heuristics of [1], it appeared that the natural hypothesis, namely not to include the p-components of the class groups for those p which divide the degree of a Galois closure, could be somewhat weakened, and we were led to the notion of a ""good prime" (cf. §2 below). Primes which do not divide the degree of the Galois closure are good, and we added some more primes to this list. In particular, 2 was considered to be a good prime for all cubic fields, though it divides the degree of the Galois closure when the field is not Galois. It was clear to us as soon as we began our work that these additional good primes could produce special difficulties [2, §3, p. 136]. However, the recent computations of G. Fung and H. Williams on nonreal cubic fields showed clearly that our recipes cannot be applied in a naive manner to the extra good primes. 2 Recall briefly some of the basic ideas of our heuristics. Let Ko be a fixed number field. We first consider (within a given algebraic closure of Q containing Ko ) a set of finite Galois extensions K/Ko with a given Galois group T (up to isomorphism) and for which the conjugacy classes of the infinite Frobenius substitutions attached to the real places of K0 are given. -
Gambling, Saving, and Lumpy Liquidity Needs Sylvan Herskowitz Online Appendix A. Additional Figures and Tables B. Sports Betting
Gambling, Saving, and Lumpy Liquidity Needs Sylvan Herskowitz Online Appendix A. Additional Figures and Tables B. Sports Betting Details B.1 Odds, Payouts, and Betting Structure B.2 Estimating the Rate of Return B.3 Characterizing Weekly Betting Profiles C. Model and Comparative Statics C.1 Demand for Gambles C.2 Increased Valuation of a Lumpy Expenditure C.3 Demand for Saving C.4 Changes in Saving Ability D. Contrasting Saving and Betting D.1 Balancing Patience and Return on Saving D.2 Estimating Return on Saving E. Additional Field Targeting Protocols 1 Appendix A: Additional Tables and Figures Figure A.1: Indirect Utility with Lumpy Good and Demand for Gambles 푉 푢 푌 − 푃 + 휂 Purchase No Purchase 푢 푌 휂 푃 푌∗ 푌 (a) Indirect utility with a lumpy good V V C E F 퐵 − 퐻 = 휎 = D + D 푚푖푛 퐵 + 푊 A + 푌 푌 ∗ ∗ 퐻 � − � � (b) Fair Gambles (c) Unfair Gambles Notes: Panel (a) shows indirect utility from income. For income levels Y > Y ∗, people will pay a price P to consume L with a utility payoff of η. Maximized utility is determined by income endowment and will be the envelope of the two pieces of the utility function. Panel (b) shows that someone with income level Y˜ will demand a fair gamble that risks reducing his income by B∗ for a chance to win W ∗ with a likelihood of winning, σ. Expected utility from the gamble is at E. Panel (c) shows that there is also demand for unfair gambles with the same loss and win amounts but win likelihood as low as σmin. -
TOWARDS the PRIME NUMBER THEOREM Contents 1. The
TOWARDS THE PRIME NUMBER THEOREM ERIC ANTLEY Abstract. Originally proven by Jacques Salomon Hadamard and Charles Jean de la Valle-Poussin independently in 1896, the prime number theorem shows the number of primes less than or equal to a given number x approaches x log x as x tends towards infinity. In this paper, a proof using complex analysis will be given to prove this statement. Contents 1. The Chebyshev Functions and Their Properties 1 2. The Riemann Zeta Function 4 3. Some Integral Transforms 12 Acknowledgments 15 References 16 Let P be the set of primes. We define the function π(x) to be the prime counting function. Definition 0.1. Given a real number x, π(x) is the the cardinality of the set fp 2 Pjp ≤ xg. Theorem 0.2 (The Prime Number Theorem). We have the following limit behav- ior: x (0.3) π(x) ∼ : log x 1. The Chebyshev Functions and Their Properties We define the function ν(x) to be the first Chebyshev function. P Definition 1.1. Given a real number x, we define ν(x) to be the sum fp2Pjp≤xg log p. We define the function (x) to be the second Chebyshev function. Definition 1.2. Given a real number x, we define (x) to be the sum over primes P p and natural numbers k of fp2Pjpk≤xg log p. We define the function Λ(x) to be the von Mangoldt function. Definition 1.3. Given a real number x, we define Λ(x) to be log p if pk = x where p is some prime and 0 else, where k 2 Z. -
ON the SELBERG-ERD˝OS PROOF of the PRIME NUMBER THEOREM 1. Introduction the Prime Numbers, and Specifically Their Enumerating F
ON THE SELBERG-ERDOS} PROOF OF THE PRIME NUMBER THEOREM ASHVIN A. SWAMINATHAN Abstract. In this article, we discuss the first elementary proof, due to Selberg and Erd}os, of the Prime Number Theorem. In particular, we begin with a presentation of the Selberg symmetry formula and proceed to give a detailed account of the proof as published by Erd}osin 1949. 1. Introduction The prime numbers, and specifically their enumerating function π defined by π(x) = #fprimes p : p ≤ xg; have been a source of fascination to mathematicians for thousands of years. As early as 300 BC, Euclid discussed a simple proof of the infinitude of primes (i.e. the unboundedness of π) in his famed Elements. As late as the 1700s, Euler discovered another, more informative proof that there are infinitely many primes: in fact, he showed that the primes are so frequent among the integers that the series X 1 p prime p diverges. Drawing on Euler's methods and employing inputs from complex analy- sis, like the study of Dirichlet series and Landau's Theorem, Dirichlet proved that there are infinitely many primes in arithmetic progressions. Such tools, and in particular the complex analytic properties of the Riemann ζ-function, were funda- mental in establishing the first asymptotic for π(x). As an adolescent in the late R x 1700s, Gauss had correctly posited that π(x) ∼ x= log x ∼ Li(x) := 2 dt= log t, but it took a century for his conjecture to be proven. In the 1840s via elementary methods, Chebyshev explicitly computed constants c1 < 1 < c2, both close to 1 in value, such that π(x) (1) c < < c ; 1 x= log x 2 a result that provided a partial validation of Gauss' conjecture.