Chapter 2 Lambert Summability and the Prime Number Theorem
|
|
- Alexia Warren
- 6 years ago
- Views:
Transcription
1 Chapter 2 Lambert Summability and the Prime Number Theorem 2. Introduction The prime number theorem (PNT) was stated as conjecture by German mathematician Carl Friedrich Gauss ( ) in the year 792 and proved independently for the first time by Jacques Hadamard and Charles Jean de la Vallée-Poussin in the same year 896. The first elementary proof of this theorem (witho using integral calculus) was given by Atle Selberg of Syracuse University in October 948. Another elementary proof of this theorem was given by Erdös in 949. The PNT describes the asymptotic distribion of the prime numbers. The PNT gives a general description of how the primes are distribed among the positive integers. Informally speaing, the PNT states that if a random integer is selected in the range of zero to some large integer N, the probability that the selected integer is prime is abo = ln.n /, where ln.n / is the natural logarithm of N. For eample, among the positive integers up to and including N D 3, abo one in seven numbers is prime, whereas up to and including N D, abo one in 23 numbers is prime (where ln.3/ D 6: and ln./ D 23:25859). In other words, the average gap between consecive prime numbers among the first N integers is roughly ln.n /: Here we give the proof of this theorem by the application of Lambert summability and Wiener s Tauberian theorem. The Lambert summability is due to German mathematician Johann Heinrich Lambert ( ) (see Hardy [4, p. 372]; Peyerimhoff [8, p. 82]; Saifi [86]). 2.2 Definitions and Notations (i) Möbius Function. The classical Möbius function.n/ is an important multiplicative function in number theory and combinatorics.this formula is due to German mathematician August Ferdinand Möbius (79 868) who M. Mursaleen, Applied Summability Methods, SpringerBriefs in Mathematics, DOI.7/ , M. Mursaleen 24 3
2 4 2 Lambert Summability and the Prime Number Theorem introduced it in 832..n/ is defined for all positive integers n and has its values in f ; ; g depending on the factorization of n into prime factors. It is defined as follows (see Peyerimhoff [8, p. 85]): 8 < ; n is a square-free positive integer with an even number of prime factors;.n/ D ;nis a square-free positive integer with an odd number of prime factors; : ; n is not square-free; that is, 8 < ; n D ;.n/ D. / : ;ndp p 2 p ;p i prime, p i p j ; ; otherwise: (2.2.) Thus (a).2/ D, since 2 D 2; (b)./ D, since D 2 5; (c).4/ D, since 4 D 2 2. We conclude that.p/ D, ifp is a prime number. (ii) The Function./. The prime-counting function./ is defined as the P number of primes not greater than, for any real number, that is,./ D p< (Peyerimhoff [8, p. 87]). For eample,./ D 4 because there are four prime numbers (2, 3, 5, and 7) less than or equal to. Similarly,./ D ;.2/ D ;.3/ D ;.4/ D 2;./ D 68;. 6 / D 78498; and. 9 / D (Hardy [43, p. 9]). (iii) The von Mangoldt Function ƒ n. The function ƒ n is defined as follows (Peryerimhoff [8, p. 84]): log p;ndp for some prime p and ; ƒ n D ; otherwise: (iv) Lambert Summability. Aseries P nd a n is said to be Lambert summable (or summable L) tos,if lim!. / a D s: (2.2.2) D In this case, we write P a n D s.l/: Note that if a series is convergent to s, then it is Lambert summable to s: This series is convergent for jj <, which is true if and only if a n D O.. C "/ n /, for every ">(see [6, 52, 99]).
3 2.3 Lemmas 5 If we write D e y.y > /; s.t/ D P t a.a D /; g.t/ D te t e t ; then P a is summable L to s if and only if (note that y ) Z lim y! y The method L is regular. te y t Z ds.t/ D lim e y t y! D lim y! y Z t s.t/dg y g t y s.t/dt D s: 2.3 Lemmas We need the following lemmas for the proof of the PNT which is stated and proved in the net section. In some cases, Tauberian condition(s) will be used to prove the required claim. The general character of a Tauberian theorem is as follows. The ordinary questions on summability consider two related sequences (or other functions) and as whether it will be true that one sequence possesses a limit whenever the other possesses a limit, the limits being the same; a Tauberian theorem appears, on the other hand, only if this is untrue, and then asserts that the one sequence possesses a limit provided the other sequence both possesses a limit and satisfies some additional condition restricting its rate of increase. The interest of a Tauberian theorem lies particularly in the character of this additional condition, which taes different forms in different cases. Lemma 2.3. (Hardy [4, p. 296]; Peyerimhoff [8, p. 8]). If g.t/; h.t/ 2 L.; /, and if Z then s.t/ D O./ (s.t/ real and measurable) and lim! Z g.t/t i dt. <</; (2.3.) t g s.t/dt D implies lim! Z t h s.t/dt D : Lemma (Peyerimhoff [8, p. 84]). If n D p p. i D ;2;:::; p i prime/; then P d=n ƒ d D log n: Proof. Since d runs through divisors of n and we have to consider only d D p ;p 2;:::;p ;:::;p ; therefore P d=n ƒ d D log p C 2 log p 2 C C log p D log n: This completes the proof of Lemma
4 6 2 Lambert Summability and the Prime Number Theorem Lemma (Peyerimhoff [8, p. 84]).. / n nd where is a Riemann s Zeta function. Proof. We have n s D. 2 s /.s/.s > /; (2.3.2). / n n s nd D 2 s C 3 s 4 s C D C 2 C 3 2 s s 2 C s 4 C s 6 C s D.s/ 2 2 s C 2 s C 3 s C D.s/ 2 s.s/ D. 2 s /.s/: This completes the proof of Lemma Lemma (Hardy [4, p. 246]). If s>, then.s/ D Y p p s p s : (2.3.3) Lemma (Hardy [43, p. 253]). Proof. From 2.3.3, wehave.s/ D.s/ nd ƒ n n s (2.3.4) log.s/ D p log p s p s : Differentiating with respect to s and observing that d p s log D log p ds p s p s ;
5 2.3 Lemmas 7 we obtain.s/.s/ D p log p p s : (2.3.5) The differentiation is legitimate because the derived series is uniformly convergent for s C ı>;ı>: We can write (2.3.5) in the form.s/.s/ D p log p md p ms and the double series PP p ms log p is absolely convergent when s>. Hence it may be written as p ms log p D ƒ n n s : p;m This completes the proof of Lemma Lemma (Peyerimhoff [8, p. 84]). s n! s.l/, asn!and a n D O n imply s n! s, asn!. Proof. We wish to show that a n D O.=n/ is a Tauberian condition. In order to apply Wiener s theory we must show that (2.3.) holds. B for "> i.e., Z Z t ic" g.t/dt D.i C "/ D.i C "/ Z D nd t ic" g.t/dt Z D.i C "/. C " C i/ t ic" e.c/t dt D. C / C"Ci t i g.t/dt D. C i/ lim "!.i C "/. C " C i/: This has a simple pole at and is on the line Rez D : A stronger theorem is true, namely, L Abel, i.e., every Lambert summable series is also Abel summable (see [42]), which implies this theorem. For the sae of completeness we give a proof that. C i/ for real. The formula (2.3.4) implies. C i/ : This completes the proof of Lemma
6 8 2 Lambert Summability and the Prime Number Theorem Lemma (Peyerimhoff [8, p. 86]). nd.n/ n D Proof. This follows from O-Tauberian theorem for Lambert summability, if D O.L/.B P nd.n/ n.n/ n. / D. /.n/ n nd D. / nd D n.c/.n/ D. /: md n=m A consequence is (by partial summation)./ D O.n/ (2.3.6) n which follows with the notation m.t/ D./ from./ D n t Z n tdm.t/ D nm.n/ Z n m.t/dt: This completes the proof of Lemma Lemma (Hardy [43, p. 346]). Suppose that c ;c 2 ;:::; is a sequence of numbers such that C.t/ D nt c n and that f.t/is any function of t. Then c n f.n/ D C.n/ff.n/ f.nc /gcc./f.œ /: (2.3.7) n n If, in addition, c j D for j<n and f.t/has a continuous derivative for t n, then Z c n f.n/ D C./f./ C.t/f.t/dt: (2.3.8) n n
7 2.3 Lemmas 9 Proof. If we write N D Œ, the sum on the left of (2.3.7)is C./f./ CfC.2/ C./gf.2/CCfC.N/ C.N /gf.n/ D C./ff./ f.2/gccc.n /ff.n / f.n/gcc.n/f.n/: Since C.N/ D C./, this proves (2.3.7). To deduce (2.3.8), we observe that C.t/ D C.n/ when n t<nc and so Z nc C.n/Œf.n/ f.nc / D n C.t/ f.t/dt: Also C.t/ D when t<n : This completes the proof of Lemma Lemma (Hardy [43, p. 347]). n where C is Euler s constant. n D log C C C O ; Proof. P c n D and f.t/d =t. WehaveC./ D Œ and (2.3.8) becomes where is independent of and n E D n D Œ Z C Œt t dt 2 D log C C C E; Z C D D Z Z D O t Œt dt t 2 t Œt Œ dt t 2 O./ dt C O t 2 This completes the proof of Lemma
8 2 2 Lambert Summability and the Prime Number Theorem Lemma 2.3. (Peyerimhoff [8, p. 86]). If./ D h C log CC i and./ D n ƒ n ; then./ D O.log. C //: Proof. Möbius formula (2.2.) yields that./ C log C C D d.d/: (2.3.9) d From log n D P d=n ƒ d Lemma 2.3.2, it follows that log n D ƒ d D ƒ d D n n ddn d= : Therefore, we obtain./ D n log n log C C C O C Œ log log C Œ C; i.e.,./ D O.log. C //: (2.3.) This completes the proof of Lemma Lemma 2.3. ([Aer s Theorem] (Peyerimhoff [8, p. 87])). If (a)./ is of bounded variation in every finite interval Œ; T ; (b) P a D O./; (c) a n D O./; (d)./ D O. / for some < <; then a D O./: Proof. Let <ı<. Then a D O. /ı D O ı : ı
9 2.3 Lemmas 2 Assuming that m <ı m, N <NC (m and N integers), we have ı a D N Dm D O./ D O./ Z ı Z =ı C ˇ ˇd ˇˇˇ C O./ t jd.t/jco./: s C s N s m N m This completes the proof of Lemma Lemma (Peyerimhoff [8, p. 87])../ D O./. Proof. It follows from (2.3.6), (2.3.9), (2.3.), and Aer s theorem, that./ D O./: This completes the proof of Lemma Lemma (Peyerimhoff [8, p. 87]). Let #./ D P p log p.pprime/; then (a) #././ D O./; (b)./ D #./ C #. p / CC#. p /, for every > log log 2. Lemma (Peyerimhoff [8, p. 87])../ D #./ C O./ log p : log 2 Proof. It follows from part (b) of Lemma that./ D #./ C O./ log p : log 2 This completes the proof of Lemma Lemma (Peyerimhoff [8, p. 87]). #./ D C O./: (2.3.) Proof. Lemma implies that #./ D C O./:
10 22 2 Lambert Summability and the Prime Number Theorem 2.4 The Prime Number Theorem Theorem The PNT states that./ is asymptotic to =log (see Hardy [4, p. 9]), that is, the limit of the quotient of the two functions./ and =ln approaches, as becomes indefinitely large, which is the same thing as Œ./ log =!, as!(peyerimhoff [8, p. 88]). Proof. By definition and by./ D Z 3=2 D #./ log C log t d#.t/ Z D #./ log C O [note that #./ D O./ : Using (2.3.) we obtain the PNT, i.e., or #.t/ t.log t/ dt 2.log / 2 3=2 lim./ D! log ;./log lim D :! This completes the proof of Theorem 2.4..
11
Chapter 1. Introduction to prime number theory. 1.1 The Prime Number Theorem
Chapter 1 Introduction to prime number theory 1.1 The Prime Number Theorem In the first part of this course, we focus on the theory of prime numbers. We use the following notation: we write f g as if lim
More informationChapter 1. Introduction to prime number theory. 1.1 The Prime Number Theorem
Chapter 1 Introduction to prime number theory 1.1 The Prime Number Theorem In the first part of this course, we focus on the theory of prime numbers. We use the following notation: we write f( g( as if
More informationTheorem 1.1 (Prime Number Theorem, Hadamard, de la Vallée Poussin, 1896). let π(x) denote the number of primes x. Then x as x. log x.
Chapter 1 Introduction 1.1 The Prime Number Theorem In this course, we focus on the theory of prime numbers. We use the following notation: we write f( g( as if lim f(/g( = 1, and denote by log the natural
More information1 Euler s idea: revisiting the infinitude of primes
8.785: Analytic Number Theory, MIT, spring 27 (K.S. Kedlaya) The prime number theorem Most of my handouts will come with exercises attached; see the web site for the due dates. (For example, these are
More informationContent. CIMPA-ICTP Research School, Nesin Mathematics Village The Riemann zeta function Michel Waldschmidt
Artin L functions, Artin primitive roots Conjecture and applications CIMPA-ICTP Research School, Nesin Mathematics Village 7 http://www.rnta.eu/nesin7/material.html The Riemann zeta function Michel Waldschmidt
More informationThe Prime Number Theorem
The Prime Number Theorem We study the distribution of primes via the function π(x) = the number of primes x 6 5 4 3 2 2 3 4 5 6 7 8 9 0 2 3 4 5 2 It s easier to draw this way: π(x) = the number of primes
More informationMATH3500 The 6th Millennium Prize Problem. The 6th Millennium Prize Problem
MATH3500 The 6th Millennium Prize Problem RH Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime
More information18.785: Analytic Number Theory, MIT, spring 2006 (K.S. Kedlaya) Dirichlet series and arithmetic functions
18.785: Analytic Number Theory, MIT, spring 2006 (K.S. Kedlaya) Dirichlet series and arithmetic functions 1 Dirichlet series The Riemann zeta function ζ is a special example of a type of series we will
More informationHarmonic sets and the harmonic prime number theorem
Harmonic sets and the harmonic prime number theorem Version: 9th September 2004 Kevin A. Broughan and Rory J. Casey University of Waikato, Hamilton, New Zealand E-mail: kab@waikato.ac.nz We restrict primes
More informationComplex Analysis: A Round-Up
Complex Analysis: A Round-Up October 1, 2009 Sergey Lototsky, USC, Dept. of Math. *** 1 Prelude: Arnold s Principle Valdimir Igorevich Arnold (b. 1937): Russian The Arnold Principle. If a notion bears
More informationExplicit Bounds Concerning Non-trivial Zeros of the Riemann Zeta Function
Explicit Bounds Concerning Non-trivial Zeros of the Riemann Zeta Function Mehdi Hassani Dedicated to Professor Hari M. Srivastava Abstract In this paper, we get explicit upper and lower bounds for n,where
More informationMath 229: Introduction to Analytic Number Theory The product formula for ξ(s) and ζ(s); vertical distribution of zeros
Math 9: Introduction to Analytic Number Theory The product formula for ξs) and ζs); vertical distribution of zeros Behavior on vertical lines. We next show that s s)ξs) is an entire function of order ;
More informationSome Fun with Divergent Series
Some Fun with Divergent Series 1. Preliminary Results We begin by examining the (divergent) infinite series S 1 = 1 + 2 + 3 + 4 + 5 + 6 + = k=1 k S 2 = 1 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 + = k=1 k 2 (i)
More information11.8 Power Series. Recall the geometric series. (1) x n = 1+x+x 2 + +x n +
11.8 1 11.8 Power Series Recall the geometric series (1) x n 1+x+x 2 + +x n + n As we saw in section 11.2, the series (1) diverges if the common ratio x > 1 and converges if x < 1. In fact, for all x (
More informationAbelian and Tauberian Theorems: Philosophy
: Philosophy Bent E. Petersen Math 515 Fall 1998 Next quarter, in Mth 516, we will discuss the Tauberian theorem of Ikehara and Landau and its connection with the prime number theorem (among other things).
More informationNotes on the Riemann Zeta Function
Notes on the Riemann Zeta Function March 9, 5 The Zeta Function. Definition and Analyticity The Riemann zeta function is defined for Re(s) > as follows: ζ(s) = n n s. The fact that this function is analytic
More informationRiemann s Zeta Function and the Prime Number Theorem
Riemann s Zeta Function and the Prime Number Theorem Dan Nichols nichols@math.umass.edu University of Massachusetts Dec. 7, 2016 Let s begin with the Basel problem, first posed in 1644 by Mengoli. Find
More informationPrime Number Theory and the Riemann Zeta-Function
5262589 - Recent Perspectives in Random Matrix Theory and Number Theory Prime Number Theory and the Riemann Zeta-Function D.R. Heath-Brown Primes An integer p N is said to be prime if p and there is no
More informationLecture 1: Small Prime Gaps: From the Riemann Zeta-Function and Pair Correlation to the Circle Method. Daniel Goldston
Lecture 1: Small Prime Gaps: From the Riemann Zeta-Function and Pair Correlation to the Circle Method Daniel Goldston π(x): The number of primes x. The prime number theorem: π(x) x log x, as x. The average
More informationTWO PROOFS OF THE PRIME NUMBER THEOREM. 1. Introduction
TWO PROOFS OF THE PRIME NUMBER THEOREM PO-LAM YUNG Introduction Let π() be the number of primes The famous prime number theorem asserts the following: Theorem (Prime number theorem) () π() log as + (This
More informationPart II. Number Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler
More informationMath 5330 Spring Notes Prime Numbers
Math 5330 Sring 208 Notes Prime Numbers The study of rime numbers is as old as mathematics itself. This set of notes has a bunch of facts about rimes, or related to rimes. Much of this stuff is old dating
More informationELEMENTARY PROOF OF DIRICHLET THEOREM
ELEMENTARY PROOF OF DIRICHLET THEOREM ZIJIAN WANG Abstract. In this expository paper, we present the Dirichlet Theorem on primes in arithmetic progressions along with an elementary proof. We first show
More informationAnalytic. Number Theory. Exploring the Anatomy of Integers. Jean-Marie. De Koninck. Florian Luca. ffk li? Graduate Studies.
Analytic Number Theory Exploring the Anatomy of Integers Jean-Marie Florian Luca De Koninck Graduate Studies in Mathematics Volume 134 ffk li? American Mathematical Society Providence, Rhode Island Preface
More informationDivergent Series: why = 1/12. Bryden Cais
Divergent Series: why + + 3 + = /. Bryden Cais Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.. H. Abel. Introduction The notion of convergence
More informationA Simple Counterexample to Havil s Reformulation of the Riemann Hypothesis
A Simple Counterexample to Havil s Reformulation of the Riemann Hypothesis Jonathan Sondow 209 West 97th Street New York, NY 0025 jsondow@alumni.princeton.edu The Riemann Hypothesis (RH) is the greatest
More information150 Years of Riemann Hypothesis.
150 Years of Riemann Hypothesis. Julio C. Andrade University of Bristol Department of Mathematics April 9, 2009 / IMPA Julio C. Andrade (University of Bristol Department 150 of Years Mathematics) of Riemann
More informationBefore giving the detailed proof, we outline our strategy. Define the functions. for Re s > 1.
Chapter 7 The Prime number theorem for arithmetic progressions 7. The Prime number theorem We denote by π( the number of primes. We prove the Prime Number Theorem. Theorem 7.. We have π( as. log Before
More informationThe Prime Number Theorem
Chapter 3 The Prime Number Theorem This chapter gives without proof the two basic results of analytic number theory. 3.1 The Theorem Recall that if f(x), g(x) are two real-valued functions, we write to
More informationThe Riemann Hypothesis
The Riemann Hypothesis Matilde N. Laĺın GAME Seminar, Special Series, History of Mathematics University of Alberta mlalin@math.ualberta.ca http://www.math.ualberta.ca/~mlalin March 5, 2008 Matilde N. Laĺın
More informationDan Buchanan. Master of Science. in the. Mathematics Program. YOUNGSTOWN STATE UNIVERSITY May, 2018
Analytic Number Theory and the Prime Number Theorem by Dan Buchanan Submitted in Partial Fulllment of the Requirements for the Degree of Master of Science in the Mathematics Program YOUNGSTOWN STATE UNIVERSITY
More informationTwin primes (seem to be) more random than primes
Twin primes (seem to be) more random than primes Richard P. Brent Australian National University and University of Newcastle 25 October 2014 Primes and twin primes Abstract Cramér s probabilistic model
More informationConversion of the Riemann prime number formula
South Asian Journal of Mathematics 202, Vol. 2 ( 5): 535 54 www.sajm-online.com ISSN 225-52 RESEARCH ARTICLE Conversion of the Riemann prime number formula Dan Liu Department of Mathematics, Chinese sichuan
More informationSmall gaps between primes
CRM, Université de Montréal Princeton/IAS Number Theory Seminar March 2014 Introduction Question What is lim inf n (p n+1 p n )? In particular, is it finite? Introduction Question What is lim inf n (p
More informationIntroduction to Analytic Number Theory Math 531 Lecture Notes, Fall 2005
Introduction to Analytic Number Theory Math 53 Lecture Notes, Fall 2005 A.J. Hildebrand Department of Mathematics University of Illinois http://www.math.uiuc.edu/~hildebr/ant Version 2005.2.06 2 Contents
More informationRiemann s insight. Where it came from and where it has led. John D. Dixon, Carleton University. November 2009
Riemann s insight Where it came from and where it has led John D. Dixon, Carleton University November 2009 JDD () Riemann hypothesis November 2009 1 / 21 BERNHARD RIEMANN Ueber die Anzahl der Primzahlen
More informationON THE DENSITY OF SOME SEQUENCES OF INTEGERS P. ERDOS
ON THE DENSITY OF SOME SEQUENCES OF INTEGERS P. ERDOS Let ai
More information. As the binomial coefficients are integers we have that. 2 n(n 1).
Math 580 Homework. 1. Divisibility. Definition 1. Let a, b be integers with a 0. Then b divides b iff there is an integer k such that b = ka. In the case we write a b. In this case we also say a is a factor
More informationON THE RESIDUE CLASSES OF π(n) MODULO t
ON THE RESIDUE CLASSES OF πn MODULO t Ping Ngai Chung Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts briancpn@mit.edu Shiyu Li 1 Department of Mathematics, University
More informationAnalysis of Selberg s Elementary Proof of the Prime Number Theorem Josue Mateo
Analysis of Selberg s Elementary Proof of the Prime Number Theorem Josue Mateo Historical Introduction Prime numbers are a concept that have intrigued mathematicians and scholars alike since the dawn of
More informationCOMPLEX ANALYSIS in NUMBER THEORY
COMPLEX ANALYSIS in NUMBER THEORY Anatoly A. Karatsuba Steklov Mathematical Institute Russian Academy of Sciences Moscow, Russia CRC Press Boca Raton Ann Arbor London Tokyo Introduction 1 Chapter 1. The
More informationArithmetic Statistics Lecture 1
Arithmetic Statistics Lecture 1 Álvaro Lozano-Robledo Department of Mathematics University of Connecticut May 28 th CTNT 2018 Connecticut Summer School in Number Theory Question What is Arithmetic Statistics?
More informationThe expansion of random regular graphs
The expansion of random regular graphs David Ellis Introduction Our aim is now to show that for any d 3, almost all d-regular graphs on {1, 2,..., n} have edge-expansion ratio at least c d d (if nd is
More informationDivisibility. 1.1 Foundations
1 Divisibility 1.1 Foundations The set 1, 2, 3,...of all natural numbers will be denoted by N. There is no need to enter here into philosophical questions concerning the existence of N. It will suffice
More informationHeuristics for Prime Statistics Brown Univ. Feb. 11, K. Conrad, UConn
Heuristics for Prime Statistics Brown Univ. Feb., 2006 K. Conrad, UConn Two quotes about prime numbers Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers,
More informationRiemann Zeta Function and Prime Number Distribution
Riemann Zeta Function and Prime Number Distribution Mingrui Xu June 2, 24 Contents Introduction 2 2 Definition of zeta function and Functional Equation 2 2. Definition and Euler Product....................
More informationTOPICS IN NUMBER THEORY - EXERCISE SHEET I. École Polytechnique Fédérale de Lausanne
TOPICS IN NUMBER THEORY - EXERCISE SHEET I École Polytechnique Fédérale de Lausanne Exercise Non-vanishing of Dirichlet L-functions on the line Rs) = ) Let q and let χ be a Dirichlet character modulo q.
More informationLinear equations in primes
ANNALS OF MATHEMATICS Linear equations in primes By Benjamin Green and Terence Tao SECOND SERIES, VOL. 171, NO. 3 May, 2010 anmaah Annals of Mathematics, 171 (2010), 1753 1850 Linear equations in primes
More informationON GENERAL PRIME NUMBER THEOREMS WITH REMAINDER
ON GENERAL PRIME NUMBER THEOREMS WITH REMAINDER GREGORY DEBRUYNE AND JASSON VINDAS Abstract. We show that for Beurling generalized numbers the prime number theorem in remainder form ( π( = Li( + O log
More informationAPPROXIMATIONS FOR SOME FUNCTIONS OF PRIMES
APPROXIMATIONS FOR SOME FUNCTIONS OF PRIMES Fataneh Esteki B. Sc., University of Isfahan, 2002 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfillment
More informationSec 4.1 Limits, Informally. When we calculated f (x), we first started with the difference quotient. f(x + h) f(x) h
1 Sec 4.1 Limits, Informally When we calculated f (x), we first started with the difference quotient f(x + h) f(x) h and made h small. In other words, f (x) is the number f(x+h) f(x) approaches as h gets
More informationA prospect proof of the Goldbach s conjecture
A prospect proof of the Goldbach s conjecture Douadi MIHOUBI LMPA, the University of M sila, 28000 M sila, Algeria mihoubi_douadi@yahoofr March 21, 2015 Abstract Based on, the well-ordering (N;
More informationON THE NUMBER OF PAIRS OF INTEGERS WITH LEAST COMMON MULTIPLE NOT EXCEEDING x H. JAGER
MATHEMATICS ON THE NUMBER OF PAIRS OF INTEGERS WITH LEAST COMMON MULTIPLE NOT EXCEEDING x BY H. JAGER (Communicated by Prof. J. PoPKEN at the meeting of March 31, 1962) In this note we shall derive an
More informationMathematical Association of America
Mathematical Association of America http://www.jstor.org/stable/2975232. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at. http://www.jstor.org/page/info/about/policies/terms.jsp
More informationDefining Prime Probability Analytically
Defining Prime Probability Analytically An elementary probability-based approach to estimating the prime and twin-prime counting functions π(n) and π 2 (n) analytically Bhupinder Singh Anand Update of
More informationFirst, let me recall the formula I want to prove. Again, ψ is the function. ψ(x) = n<x
8.785: Analytic Number heory, MI, spring 007 (K.S. Kedlaya) von Mangoldt s formula In this unit, we derive von Mangoldt s formula estimating ψ(x) x in terms of the critical zeroes of the Riemann zeta function.
More informationNeedles and Numbers. The Buffon Needle Experiment
eedles and umbers This excursion into analytic number theory is intended to complement the approach of our textbook, which emphasizes the algebraic theory of numbers. At some points, our presentation lacks
More informationElementary Number Theory
Elementary Number Theory 21.8.2013 Overview The course discusses properties of numbers, the most basic mathematical objects. We are going to follow the book: David Burton: Elementary Number Theory What
More informationSummation Methods on Divergent Series. Nick Saal Santa Rosa Junior College April 23, 2016
Summation Methods on Divergent Series Nick Saal Santa Rosa Junior College April 23, 2016 Infinite series are incredibly useful tools in many areas of both pure and applied mathematics, as well as the sciences,
More informationEuler and the modern mathematician. V. S. Varadarajan Department of Mathematics, UCLA. University of Bologna
Euler and the modern mathematician V. S. Varadarajan Department of Mathematics, UCLA University of Bologna September 2008 Abstract It is remarkable that Euler, who preceded Gauss and Riemann, is still
More informationPATTERNS OF PRIMES IN ARITHMETIC PROGRESSIONS
PATTERNS OF PRIMES IN ARITHMETIC PROGRESSIONS JÁNOS PINTZ Rényi Institute of the Hungarian Academy of Sciences CIRM, Dec. 13, 2016 2 1. Patterns of primes Notation: p n the n th prime, P = {p i } i=1,
More informationThe Riemann Hypothesis
The Riemann Hypothesis by Pedro Pablo Pérez Velasco, 2011 Introduction Consider the series of questions: How many prime numbers (2, 3, 5, 7,...) are there less than 100, less than 10.000, less than 1.000.000?
More informationDirichlet s Theorem. Martin Orr. August 21, The aim of this article is to prove Dirichlet s theorem on primes in arithmetic progressions:
Dirichlet s Theorem Martin Orr August 1, 009 1 Introduction The aim of this article is to prove Dirichlet s theorem on primes in arithmetic progressions: Theorem 1.1. If m, a N are coprime, then there
More informationHere is another characterization of prime numbers.
Here is another characterization of prime numbers. Theorem p is prime it has no divisors d that satisfy < d p. Proof [ ] If p is prime then it has no divisors d that satisfy < d < p, so clearly no divisor
More informationGod may not play dice with the universe, but something strange is going on with the prime numbers.
Primes: Definitions God may not play dice with the universe, but something strange is going on with the prime numbers. P. Erdös (attributed by Carl Pomerance) Def: A prime integer is a number whose only
More informationThe zeta function, L-functions, and irreducible polynomials
The zeta function, L-functions, and irreducible polynomials Topics in Finite Fields Fall 203) Rutgers University Swastik Kopparty Last modified: Sunday 3 th October, 203 The zeta function and irreducible
More informationPrimes and Factorization
Primes and Factorization 1 A prime number is an integer greater than 1 with no proper divisors. The list begins 2, 3, 5, 7, 11, 13, 19,... See http://primes.utm.edu/ for a wealth of information about primes.
More informationSJÄLVSTÄNDIGA ARBETEN I MATEMATIK
SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Gamma function related to Pic functions av Saad Abed 25 - No 4 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 6 9
More informationTaylor Series and Asymptotic Expansions
Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.
More informationMath 229: Introduction to Analytic Number Theory Čebyšev (and von Mangoldt and Stirling)
ath 9: Introduction to Analytic Number Theory Čebyšev (and von angoldt and Stirling) Before investigating ζ(s) and L(s, χ) as functions of a complex variable, we give another elementary approach to estimating
More informationZeros of ζ (s) & ζ (s) in σ< 1 2
Turk J Math 4 (000, 89 08. c TÜBİTAK Zeros of (s & (s in σ< Cem Yalçın Yıldırım Abstract There is only one pair of non-real zeros of (s, and of (s, in the left halfplane. The Riemann Hypothesis implies
More informationDirichlet s Theorem. Calvin Lin Zhiwei. August 18, 2007
Dirichlet s Theorem Calvin Lin Zhiwei August 8, 2007 Abstract This paper provides a proof of Dirichlet s theorem, which states that when (m, a) =, there are infinitely many primes uch that p a (mod m).
More informationVII.8. The Riemann Zeta Function.
VII.8. The Riemann Zeta Function VII.8. The Riemann Zeta Function. Note. In this section, we define the Riemann zeta function and discuss its history. We relate this meromorphic function with a simple
More informationMathematics 324 Riemann Zeta Function August 5, 2005
Mathematics 324 Riemann Zeta Function August 5, 25 In this note we give an introduction to the Riemann zeta function, which connects the ideas of real analysis with the arithmetic of the integers. Define
More informationPrime and Perfect Numbers
Prime and Perfect Numbers 0.3 Infinitude of prime numbers 0.3.1 Euclid s proof Euclid IX.20 demonstrates the infinitude of prime numbers. 1 The prime numbers or primes are the numbers 2, 3, 5, 7, 11, 13,
More informationζ (s) = s 1 s {u} [u] ζ (s) = s 0 u 1+sdu, {u} Note how the integral runs from 0 and not 1.
Problem Sheet 3. From Theorem 3. we have ζ (s) = + s s {u} u+sdu, (45) valid for Res > 0. i) Deduce that for Res >. [u] ζ (s) = s u +sdu ote the integral contains [u] in place of {u}. ii) Deduce that for
More informationThe Convolution Square Root of 1
The Convolution Square Root of 1 Harold G. Diamond University of Illinois, Urbana. AMS Special Session in Honor of Jeff Vaaler January 12, 2018 0. Sections of the talk 1. 1 1/2, its convolution inverse
More informationWeek 2: Sequences and Series
QF0: Quantitative Finance August 29, 207 Week 2: Sequences and Series Facilitator: Christopher Ting AY 207/208 Mathematicians have tried in vain to this day to discover some order in the sequence of prime
More informationMATH10040: Chapter 0 Mathematics, Logic and Reasoning
MATH10040: Chapter 0 Mathematics, Logic and Reasoning 1. What is Mathematics? There is no definitive answer to this question. 1 Indeed, the answer given by a 21st-century mathematician would differ greatly
More informationGauss and Riemann versus elementary mathematics
777-855 826-866 Gauss and Riemann versus elementary mathematics Problem at the 987 International Mathematical Olympiad: Given that the polynomial [ ] f (x) = x 2 + x + p yields primes for x =,, 2,...,
More informationFunctions of a Complex Variable
MIT OenCourseWare htt://ocw.mit.edu 8. Functions of a Comle Variable Fall 8 For information about citing these materials or our Terms of Use, visit: htt://ocw.mit.edu/terms. Lecture and : The Prime Number
More informationA Painless Overview of the Riemann Hypothesis
A Painless Overview of the Riemann Hypothesis [Proof Omitted] Peter Lynch School of Mathematics & Statistics University College Dublin Irish Mathematical Society 40 th Anniversary Meeting 20 December 2016.
More informationA Painless Overview of the Riemann Hypothesis
A Painless Overview of the Riemann Hypothesis [Proof Omitted] Peter Lynch School of Mathematics & Statistics University College Dublin Irish Mathematical Society 40 th Anniversary Meeting 20 December 2016.
More informationCHAPTER 4. Series. 1. What is a Series?
CHAPTER 4 Series Given a sequence, in many contexts it is natural to ask about the sum of all the numbers in the sequence. If only a finite number of the are nonzero, this is trivial and not very interesting.
More information11.6: Ratio and Root Tests Page 1. absolutely convergent, conditionally convergent, or divergent?
.6: Ratio and Root Tests Page Questions ( 3) n n 3 ( 3) n ( ) n 5 + n ( ) n e n ( ) n+ n2 2 n Example Show that ( ) n n ln n ( n 2 ) n + 2n 2 + converges for all x. Deduce that = 0 for all x. Solutions
More informationIndustrial Strength Factorization. Lawren Smithline Cornell University
Industrial Strength Factorization Lawren Smithline Cornell University lawren@math.cornell.edu http://www.math.cornell.edu/~lawren Industrial Strength Factorization Given an integer N, determine the prime
More informationVerbatim copying and redistribution of this document. are permitted in any medium provided this notice and. the copyright notice are preserved.
New Results on Primes from an Old Proof of Euler s by Charles W. Neville CWN Research 55 Maplewood Ave. West Hartford, CT 06119, U.S.A. cwneville@cwnresearch.com September 25, 2002 Revision 1, April, 2003
More informationFormulae for Computing Logarithmic Integral Function ( )!
Formulae for Computing Logarithmic Integral Function x 2 ln t Li(x) dt Amrik Singh Nimbran 6, Polo Road, Patna, INDIA Email: simnimas@yahoo.co.in Abstract: The prime number theorem states that the number
More informationProbabilistic Aspects of the Integer-Polynomial Analogy
Probabilistic Aspects of the Integer-Polynomial Analogy Kent E. Morrison Department of Mathematics California Polytechnic State University San Luis Obispo, CA 93407 kmorriso@calpoly.edu Zhou Dong Department
More informationFormalizing an Analytic Proof of the Prime Number Theorem Dedicated to Mike Gordon
0 Formalizing an Analytic Proof of the Prime Number Theorem Dedicated to Mike Gordon John Harrison Intel Corporation TTVSI (Mikefest) Royal Society, London Wed 26th March 2008 (10:00 10:45) 1 What is the
More informationA proof of Selberg s orthogonality for automorphic L-functions
A proof of Selberg s orthogonality for automorphic L-functions Jianya Liu, Yonghui Wang 2, and Yangbo Ye 3 Abstract Let π and π be automorphic irreducible cuspidal representations of GL m and GL m, respectively.
More informationMath 680 Fall A Quantitative Prime Number Theorem I: Zero-Free Regions
Math 68 Fall 4 A Quantitative Prime Number Theorem I: Zero-Free Regions Ultimately, our goal is to prove the following strengthening of the prime number theorem Theorem Improved Prime Number Theorem: There
More informationEvidence for the Riemann Hypothesis
Evidence for the Riemann Hypothesis Léo Agélas September 0, 014 Abstract Riemann Hypothesis (that all non-trivial zeros of the zeta function have real part one-half) is arguably the most important unsolved
More informationWe start with a simple result from Fourier analysis. Given a function f : [0, 1] C, we define the Fourier coefficients of f by
Chapter 9 The functional equation for the Riemann zeta function We will eventually deduce a functional equation, relating ζ(s to ζ( s. There are various methods to derive this functional equation, see
More informationPrimes go Quantum: there is entanglement in the primes
Primes go Quantum: there is entanglement in the primes Germán Sierra In collaboration with José Ignacio Latorre Madrid-Barcelona-Singapore Workshop: Entanglement Entropy in Quantum Many-Body Systems King's
More informationOne of the Eight Numbers ζ(5),ζ(7),...,ζ(17),ζ(19) Is Irrational
Mathematical Notes, vol. 70, no. 3, 2001, pp. 426 431. Translated from Matematicheskie Zametki, vol. 70, no. 3, 2001, pp. 472 476. Original Russian Text Copyright c 2001 by V. V. Zudilin. One of the Eight
More informationPolynomial Formula for Sums of Powers of Integers
Polynomial Formula for Sums of Powers of Integers K B Athreya and S Kothari left K B Athreya is a retired Professor of mathematics and statistics at Iowa State University, Ames, Iowa. right S Kothari is
More informationPretentiousness in analytic number theory. Andrew Granville A report on joint work with K. Soundararajan, and Antal Balog
Pretentiousness in analytic number theory Andrew Granville A report on joint work with K. Soundararajan, and Antal Balog The number of primes up to x Gauss, Christmas eve 1849: As a boy of 15 or 16, I
More informationNOTES ON RIEMANN S ZETA FUNCTION. Γ(z) = t z 1 e t dt
NOTES ON RIEMANN S ZETA FUNCTION DRAGAN MILIČIĆ. Gamma function.. Definition of the Gamma function. The integral Γz = t z e t dt is well-defined and defines a holomorphic function in the right half-plane
More informationMTH598A Report The Vinogradov Theorem
MTH598A Report The Vinogradov Theorem Anurag Sahay 11141/11917141 under the supervision of Dr. Somnath Jha Dept. of Mathematics and Statistics 4th November, 2015 Abstract The Goldbach conjecture is one
More information