Probability and Number Theory: an Overview of the Erdős-Kac Theorem
|
|
- Blaze Arnold
- 6 years ago
- Views:
Transcription
1 Probability and umber Theory: an Overview of the Erdős-Kac Theorem December 12, 2012 Probability and umber Theory: an Overview of the Erdős-Kac Theorem
2 Probabilistic number theory? Pick n with n 10, 000, 000 at random. How likely is it to be prime? How many prime divisors will it have? Probability and umber Theory: an Overview of the Erdős-Kac Theorem
3 The prime divisor counting function ω Definition The function ω : defined by ω(n) := {p:p n} is called the prime divisor counting function; ω(n) yields the number of distinct prime divisors of n. 1 Probability and umber Theory: an Overview of the Erdős-Kac Theorem
4 The prime divisor counting function ω Definition The function ω : defined by ω(n) := {p:p n} is called the prime divisor counting function; ω(n) yields the number of distinct prime divisors of n. n prime factorization ω(n) Probability and umber Theory: an Overview of the Erdős-Kac Theorem
5 The prime divisor counting function ω Definition The function ω : defined by ω(n) := {p:p n} is called the prime divisor counting function; ω(n) yields the number of distinct prime divisors of n. n prime factorization ω(n) Probability and umber Theory: an Overview of the Erdős-Kac Theorem
6 The prime divisor counting function ω Definition The function ω : defined by ω(n) := {p:p n} is called the prime divisor counting function; ω(n) yields the number of distinct prime divisors of n. n prime factorization ω(n) Probability and umber Theory: an Overview of the Erdős-Kac Theorem
7 The prime divisor counting function ω Definition The function ω : defined by ω(n) := {p:p n} is called the prime divisor counting function; ω(n) yields the number of distinct prime divisors of n. n prime factorization ω(n) Probability and umber Theory: an Overview of the Erdős-Kac Theorem
8 An Illustration ω(n) n Probability and umber Theory: an Overview of the Erdős-Kac Theorem
9 An Illustration 1 2 3
10 An Illustration # {n : ω(n) } = Probability and umber Theory: an Overview of the Erdős-Kac Theorem
11 The Erdős-Kac Theorem Theorem Let. Then as, { ω(n) log log ν n : log log } = Φ(). That is, the limit distribution of the prime-divisor counting function ω(n) is the normal distribution with mean log log and variance log log. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
12 An Illustration Probability and umber Theory: an Overview of the Erdős-Kac Theorem
13 # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } An Illustration = 10 = 20 = 30 = 40 = 50 = = 5000 = 1000 = 500 = 100 = = = = Probability and umber Theory: an Overview of the Erdős-Kac Theorem
14 The Erdős-Kac Theorem Heuristically: 1. Most numbers near a fied have log log prime factors (Hardy and Ramanujan, Turán). Probability and umber Theory: an Overview of the Erdős-Kac Theorem
15 The Erdős-Kac Theorem Heuristically: 1. Most numbers near a fied have log log prime factors (Hardy and Ramanujan, Turán). 2. Most prime factors of most numbers near are small. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
16 The Erdős-Kac Theorem Heuristically: 1. Most numbers near a fied have log log prime factors (Hardy and Ramanujan, Turán). 2. Most prime factors of most numbers near are small. 3. The events p divides n, with p a small prime, are roughly independent (Brun sieve). Probability and umber Theory: an Overview of the Erdős-Kac Theorem
17 The Erdős-Kac Theorem Heuristically: 1. Most numbers near a fied have log log prime factors (Hardy and Ramanujan, Turán). 2. Most prime factors of most numbers near are small. 3. The events p divides n, with p a small prime, are roughly independent (Brun sieve). 4. If the events were eactly independent, a normal distribution would result. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
18 Erdős-Kac vs. Central Limit Theorem Theorem Let X 1, X 2,... be a sequence of independent and identically distributed random variables, each having mean µ and variance σ 2. Then the distribution of X X n nµ σ n tends to the standard normal as n. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
19 Uniform probability law By ν we denote the probability law of the uniform distribution with weight 1 on {1, 2,..., }. That is, for A, ν A = { 1 λ n with λ = n 0 n >. n A Probability and umber Theory: an Overview of the Erdős-Kac Theorem
20 Weak convergence We say that a sequence {F n } of distribution functions converges weakly to a function F if lim F n() = F () n for all points where F is continuous. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
21 Limiting distributions Let f be an arithmetic function. Let. Define F (z) := ν {n : f (n) z} = 1 # {n : f (n) z}. We say that f possess a limiting distribution function F if the sequence F converges weakly to a limit F that is a distribution function. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
22 # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } Limiting distributions Let f be an arithmetic function. Let. Define F (z) := ν {n : f (n) z} = 1 # {n : f (n) z}. We say that f possess a limiting distribution function F if the sequence F converges weakly to a limit F that is a distribution function. = 10 = 50 = 500 = 5000 = Probability and umber Theory: an Overview of the Erdős-Kac Theorem
23 Characteristic functions Definition Let F be a distribution function. Then its characteristic function is given by ϕ F (τ) := ep(iτ z)df (z). The characteristic function is uniformly continuous on the real line. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
24 Characteristic functions Definition Let F be a distribution function. Then its characteristic function is given by ϕ F (τ) := ep(iτ z)df (z). The characteristic function is uniformly continuous on the real line. Fact: A distribution function is completely characterized by its characteristic function. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
25 Characteristic functions Definition Let F be a distribution function. Then its characteristic function is given by ϕ F (τ) := ep(iτ z)df (z). The characteristic function is uniformly continuous on the real line. Fact: A distribution function is completely characterized by its characteristic function. Lemma The characteristic function of the standard normal distribution Φ is given by ϕ Φ (τ) = ep ( τ 2 ). 2 Probability and umber Theory: an Overview of the Erdős-Kac Theorem
26 Levy s continuity theorem Theorem Let {F n } be a sequence of distribution functions and {ϕ Fn } be the corresponding sequence of their characteristic functions. Then {F n } converges weakly to a distribution function F if and only if ϕ Fn converges pointwise on R to a function ϕ that is continuous at 0. Additionally, in this case, ϕ is the characteristic function of F. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
27 The Erdős-Kac Theorem Theorem Let. Then as, { ω(n) log log ν n : log log } = Φ(). That is, the limit distribution of the prime-divisor counting function ω(n) is the normal distribution with mean log log and variance log log. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
28 A proof sketch The atomic distribution function for is { ω(n) log log F () = ν n : log log = 1 { # ω(n) log log n : log log } }. We denote by ϕ F (τ) the characteristic function of F. We have ϕ F (τ) = e iτz df (z) Let P = { < 1 < 0 < 1 < < i } be a partition of R. Then we have ϕ F (τ) equal to Probability and umber Theory: an Overview of the Erdős-Kac Theorem
29 A proof sketch continued = = lim mesh(p)0 = lim mesh(p)0 [ = 1 = 1 = 1 e iτz df (z) e iτz (F ( k ) F ( k 1 )) k k lim mesh(p)0 ( 1 e iτz # {n : f (n) k } 1 ) # {n : f (n) k 1 } e iτz (# {n : f (n) k } # {n : f (n) k 1 } )] k ma{ω(n):n } n k=0 iτf (n) e iτf (n) e Probability and umber Theory: an Overview of the Erdős-Kac Theorem
30 A proof sketch continued et, we find some bounds for ϕ F (τ): ϕ F (τ) = ep ( τ 2 ) ( ( )) ( ) τ + τ O + O. 2 log log log (Informally, we write f () = O(g()) when there eists a positive function g such that f does not grow faster than g.) Probability and umber Theory: an Overview of the Erdős-Kac Theorem
31 A proof sketch continued et, we find some bounds for ϕ F (τ): ϕ F (τ) = ep ( τ 2 ) ( ( )) ( ) τ + τ O + O. 2 log log log (Informally, we write f () = O(g()) when there eists a positive function g such that f does not grow faster than g.) Take the limit as : ϕ F (τ) ep ( τ 2 ) = ϕ Φ (τ). 2 Probability and umber Theory: an Overview of the Erdős-Kac Theorem
32 A proof sketch continued In other words, the sequence of characteristic functions ϕ F converges pointwise to the characteristic function of the normal distribution. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
33 A proof sketch continued In other words, the sequence of characteristic functions ϕ F converges pointwise to the characteristic function of the normal distribution. Apply Levy s continuity theorem: { ω(n) log log ν n : log log } = Φ(). Thus, the limit distribution of the prime-divisor counting function ω(n) is the normal distribution with mean log log and variance log log. This completes the proof. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
34 # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } An Illustration = 10 = 20 = 30 = 40 = 50 = = 5000 = 1000 = 500 = 100 = = = = Probability and umber Theory: an Overview of the Erdős-Kac Theorem
35 References Steuding, J. Probabilistic umber Theory Gowers, T. The Importance of Mathematics. Probability and umber Theory: an Overview of the Erdős-Kac Theorem
36 Probability and umber Theory: an Overview of the Erdős-Kac Theorem
37 Probability and umber Theory: an Overview of the Erdős-Kac Theorem
Counting subgroups of the multiplicative group
Counting subgroups of the multiplicative group Lee Troupe joint w/ Greg Martin University of British Columbia West Coast Number Theory 2017 Question from I. Shparlinski to G. Martin, circa 2009: How many
More informationA 1935 Erdős paper on prime numbers and Euler s function
A 1935 Erdős paper on prime numbers and Euler s function Carl Pomerance, Dartmouth College with Florian Luca, UNAM, Morelia 1 2 3 4 Hardy & Ramanujan, 1917: The normal number of prime divisors of n is
More informationOn the number of co-prime-free sets. Neil J. Calkin and Andrew Granville *
On the number of co-prime-free sets. by Neil J. Calkin and Andrew Granville * Abstract: For a variety of arithmetic properties P such as the one in the title) we investigate the number of subsets of the
More informationOn the average number of divisors of the Euler function
On the average number of divisors of the Euler function Florian Luca Instituto de Matemáticas Universidad Nacional Autonoma de Méico CP 58089, Morelia, Michoacán, Méico fluca@matmorunamm Carl Pomerance
More informationPALINDROMIC SUMS OF PROPER DIVISORS. Paul Pollack Department of Mathematics, University of Georgia, Athens, GA 30602, USA
PALINDROMIC SUMS OF PROPER DIVISORS Paul Pollack Department of Mathematics, University of Georgia, Athens, GA 30602, USA pollack@math.uga.edu Received:, Revised:, Accepted:, Published: Abstract Fi an integer
More informationAliquot sums of Fibonacci numbers
Aliquot sums of Fibonacci numbers Florian Luca Instituto de Matemáticas Universidad Nacional Autónoma de Méico C.P. 58089, Morelia, Michoacán, Méico fluca@matmor.unam.m Pantelimon Stănică Naval Postgraduate
More informationFlat primes and thin primes
Flat primes and thin primes Kevin A. Broughan and Zhou Qizhi University of Waikato, Hamilton, New Zealand Version: 0th October 2008 E-mail: kab@waikato.ac.nz, qz49@waikato.ac.nz Flat primes and thin primes
More informationDistinct edge weights on graphs
of Distinct edge weights on University of California-San Diego mtait@math.ucsd.edu Joint work with Jacques Verstraëte November 4, 2014 (UCSD) of November 4, 2014 1 / 46 Overview of 1 2 3 Product-injective
More informationarxiv: v1 [math.co] 24 Apr 2014
On sets of integers with restrictions on their products Michael Tait, Jacques Verstraëte Department of Mathematics University of California at San Diego 9500 Gilman Drive, La Jolla, California 9093-011,
More informationArithmetic progressions in primes
Arithmetic progressions in primes Alex Gorodnik CalTech Lake Arrowhead, September 2006 Introduction Theorem (Green,Tao) Let A P := {primes} such that Then for any k N, for some a, d N. lim sup N A [1,
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 informationTHE CONVEX HULL OF THE PRIME NUMBER GRAPH
THE CONVEX HULL OF THE PRIME NUMBER GRAPH NATHAN MCNEW Abstract Let p n denote the n-th prime number, and consider the prime number graph, the collection of points n, p n in the plane Pomerance uses the
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 informationNotes on the second moment method, Erdős multiplication tables
Notes on the second moment method, Erdős multiplication tables January 25, 20 Erdős multiplication table theorem Suppose we form the N N multiplication table, containing all the N 2 products ab, where
More informationSums of distinct divisors
Oberlin College February 3, 2017 1 / 50 The anatomy of integers 2 / 50 What does it mean to study the anatomy of integers? Some natural problems/goals: Study the prime factors of integers, their size and
More informationEuler s ϕ function. Carl Pomerance Dartmouth College
Euler s ϕ function Carl Pomerance Dartmouth College Euler s ϕ function: ϕ(n) is the number of integers m [1, n] with m coprime to n. Or, it is the order of the unit group of the ring Z/nZ. Euler: If a
More informationThe Green-Tao Theorem on arithmetic progressions within the primes. Thomas Bloom
The Green-Tao Theorem on arithmetic progressions within the primes Thomas Bloom November 7, 2010 Contents 1 Introduction 6 1.1 Heuristics..................................... 7 1.2 Arithmetic Progressions.............................
More informationPALINDROMIC SUMS OF PROPER DIVISORS. Paul Pollack Department of Mathematics, University of Georgia, Athens, Georgia
#A3 INTEGERS 5A (205) PALINDROMIC SUMS OF PROPER DIVISORS Paul Pollack Department of Mathematics, University of Georgia, Athens, Georgia pollack@math.uga.edu Received: 0/29/3, Accepted: 6/8/4, Published:
More informationCompositions with the Euler and Carmichael Functions
Compositions with the Euler and Carmichael Functions William D. Banks Department of Mathematics University of Missouri Columbia, MO 652, USA bbanks@math.missouri.edu Florian Luca Instituto de Matemáticas
More informationRepeated Values of Euler s Function. a talk by Paul Kinlaw on joint work with Jonathan Bayless
Repeated Values of Euler s Function a talk by Paul Kinlaw on joint work with Jonathan Bayless Oct 4 205 Problem of Erdős: Consider solutions of the equations ϕ(n) = ϕ(n + k), σ(n) = σ(n + k) for fixed
More informationStatistics in elementary number theory
Statistics in elementary number theory Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA CRM Workshop on Statistics and Number Theory 15 19 September, 2014 In this talk we will consider 4 statistical
More informationAlgorithms for the Multiplication Table Problem
Algorithms for the Multiplication Table Problem Richard P. Brent Australian National University and CARMA, University of Newcastle 29 May 2018 Collaborators Carl Pomerance and Jonathan Webster Copyright
More informationHOW OFTEN IS EULER S TOTIENT A PERFECT POWER? 1. Introduction
HOW OFTEN IS EULER S TOTIENT A PERFECT POWER? PAUL POLLACK Abstract. Fix an integer k 2. We investigate the number of n x for which ϕn) is a perfect kth power. If we assume plausible conjectures on the
More informationMath 104: Homework 7 solutions
Math 04: Homework 7 solutions. (a) The derivative of f () = is f () = 2 which is unbounded as 0. Since f () is continuous on [0, ], it is uniformly continous on this interval by Theorem 9.2. Hence for
More informationGeneralized multiplication tables
Generalized multiplication tables Dimitris Koukoulopoulos Centre de Recherches Mathématiques October 3, 2010 Dimitris Koukoulopoulos (CRM) Generalized multiplication tables October 3, 2010 1 / 16 The Erdős
More informationITERATES OF THE SUM OF THE UNITARY DIVISORS OF AN INTEGER
Annales Univ. Sci. Budapest., Sect. Comp. 45 (06) 0 0 ITERATES OF THE SUM OF THE UNITARY DIVISORS OF AN INTEGER Jean-Marie De Koninck (Québec, Canada) Imre Kátai (Budapest, Hungary) Dedicated to Professor
More informationPartitions into Values of a Polynomial
Partitions into Values of a Polynomial Ayla Gafni University of Rochester Connections for Women: Analytic Number Theory Mathematical Sciences Research Institute February 2, 2017 Partitions A partition
More informationOn the maximal exponent of the prime power divisor of integers
Acta Univ. Sapientiae, Mathematica, 7, 205 27 34 DOI: 0.55/ausm-205-0003 On the maimal eponent of the prime power divisor of integers Imre Kátai Faculty of Informatics Eötvös Loránd University Budapest,
More informationEötvös Loránd University Faculty of Informatics. Distribution of additive arithmetical functions
Eötvös Loránd University Faculty of Informatics Distribution of additive arithmetical functions Theses of Ph.D. Dissertation by László Germán Suervisor Prof. Dr. Imre Kátai member of the Hungarian Academy
More informationON THE UNIFORM DISTRIBUTION OF CERTAIN SEQUENCES INVOLVING THE EULER TOTIENT FUNCTION AND THE SUM OF DIVISORS FUNCTION
Annales Univ. Sci. Budapest., Sect. Comp. 44 (205) 79 9 ON THE UNIFORM DISTRIBUTION OF CERTAIN SEQUENCES INVOLVING THE EULER TOTIENT FUNCTION AND THE SUM OF DIVISORS FUNCTION Jean-Marie De Koninck (Québec,
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 informationOn Erdős and Sárközy s sequences with Property P
Monatsh Math 017 18:565 575 DOI 10.1007/s00605-016-0995-9 On Erdős and Sárközy s sequences with Proerty P Christian Elsholtz 1 Stefan Planitzer 1 Received: 7 November 015 / Acceted: 7 October 016 / Published
More informationAnalytic number theory for probabilists
Analytic number theory for probabilists E. Kowalski ETH Zürich 27 October 2008 Je crois que je l ai su tout de suite : je partirais sur le Zéta, ce serait mon navire Argo, celui qui me conduirait à la
More informationTHE DISTRIBUTION OF ADDITIVE FUNCTIONS IN SHORT INTERVALS ON THE SET OF SHIFTED INTEGERS HAVING A FIXED NUMBER OF PRIME FACTORS
Annales Univ. Sci. Budaest., Sect. Com. 38 202) 57-70 THE DISTRIBUTION OF ADDITIVE FUNCTIONS IN SHORT INTERVALS ON THE SET OF SHIFTED INTEGERS HAVING A FIXED NUMBER OF PRIME FACTORS J.-M. De Koninck Québec,
More informationx x 1 0 1/(N 1) (N 2)/(N 1)
Please simplify your answers to the extent reasonable without a calculator, show your work, and explain your answers, concisely. If you set up an integral or a sum that you cannot evaluate, leave it as
More informationOn rational numbers associated with arithmetic functions evaluated at factorials
On rational numbers associated with arithmetic functions evaluated at factorials Dan Baczkowski (joint work with M. Filaseta, F. Luca, and O. Trifonov) (F. Luca) Fix r Q, there are a finite number of positive
More informationArithmetic Functions Evaluated at Factorials!
Arithmetic Functions Evaluated at Factorials! Dan Baczkowski (joint work with M. Filaseta, F. Luca, and O. Trifonov) (F. Luca) Fix r Q, there are a finite number of positive integers n and m for which
More informationSome mysteries of multiplication, and how to generate random factored integers
Some mysteries of multiplication, and how to generate random factored integers Richard P. Brent Australian National University 6 February 2015 joint work with Carl Pomerance Presented at Hong Kong Baptist
More informationA remark on a conjecture of Chowla
J. Ramanujan Math. Soc. 33, No.2 (2018) 111 123 A remark on a conjecture of Chowla M. Ram Murty 1 and Akshaa Vatwani 2 1 Department of Mathematics, Queen s University, Kingston, Ontario K7L 3N6, Canada
More informationROTH S THEOREM THE FOURIER ANALYTIC APPROACH NEIL LYALL
ROTH S THEOREM THE FOURIER AALYTIC APPROACH EIL LYALL Roth s Theorem. Let δ > 0 and ep ep(cδ ) for some absolute constant C. Then any A {0,,..., } of size A = δ necessarily contains a (non-trivial) arithmetic
More informationLecture 3: September 10
CS294 Markov Chain Monte Carlo: Foundations & Applications Fall 2009 Lecture 3: September 10 Lecturer: Prof. Alistair Sinclair Scribes: Andrew H. Chan, Piyush Srivastava Disclaimer: These notes have not
More informationJEAN-MARIE DE KONINCK AND IMRE KÁTAI
BULLETIN OF THE HELLENIC MATHEMATICAL SOCIETY Volume 6, 207 ( 0) ON THE DISTRIBUTION OF THE DIFFERENCE OF SOME ARITHMETIC FUNCTIONS JEAN-MARIE DE KONINCK AND IMRE KÁTAI Abstract. Let ϕ stand for the Euler
More information. In particular if a b then N(
Gaussian Integers II Let us summarise what we now about Gaussian integers so far: If a, b Z[ i], then N( ab) N( a) N( b). In particular if a b then N( a ) N( b). Let z Z[i]. If N( z ) is an integer prime,
More informationON THE SUM OF DIVISORS FUNCTION
Annales Univ. Sci. Budapest., Sect. Comp. 40 2013) 129 134 ON THE SUM OF DIVISORS FUNCTION N.L. Bassily Cairo, Egypt) Dedicated to Professors Zoltán Daróczy and Imre Kátai on their 75th anniversary Communicated
More informationSOME RESULTS AND PROBLEMS IN PROBABILISTIC NUMBER THEORY
Annales Univ. Sci. Budapest., Sect. Comp. 43 204 253 265 SOME RESULTS AND PROBLEMS IN PROBABILISTIC NUMBER THEORY Imre Kátai and Bui Minh Phong Budapest, Hungary Le Manh Thanh Hue, Vietnam Communicated
More informationO N P O S I T I V E N U M B E R S n F O R W H I C H Q(n) D I V I D E S F n
O N P O S I T I V E N U M B E R S n F O R W H I C H Q(n) D I V I D E S F n Florian Luca IMATE de la UNAM, Ap. Postal 61-3 (Xangari), CP 58 089, Morelia, Michoacan, Mexico e-mail: fluca@matmor.unain.inx
More informationThe first dynamical system
The first dynamical system Carl Pomerance, Dartmouth College Charles University November 8, 2016 As we all know, functions in mathematics are ubiquitous and indispensable. But what was the very first function
More informationPolynomials over finite fields. Algorithms and Randomness
Polynomials over Finite Fields: Algorithms and Randomness School of Mathematics and Statistics Carleton University daniel@math.carleton.ca AofA 10, July 2010 Introduction Let q be a prime power. In this
More informationThe 98th Mathematics Colloquium Prague November 8, Random number theory. Carl Pomerance, Dartmouth College
The 98th Mathematics Colloquium Prague November 8, 2016 Random number theory Carl Pomerance, Dartmouth College In 1770, Euler wrote: Mathematicians have tried in vain to discover some order in the sequence
More informationCONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ω(q) AND ν(q)
CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE Abstract. Recently, Andrews, Dixit and Yee introduced partition
More informationON THE DENSITY OF SOME SEQUENCES OF INTEGERS P. ERDOS
ON THE DENSITY OF SOME SEQUENCES OF INTEGERS P. ERDOS Let ai
More informationCounting Prime Numbers with Short Binary Signed Representation
Counting Prime Numbers with Short Binary Signed Representation José de Jesús Angel Angel and Guillermo Morales-Luna Computer Science Section, CINVESTAV-IPN, Mexico jjangel@computacion.cs.cinvestav.mx,
More informationPrimitive roots in algebraic number fields
Primitive roots in algebraic number fields Research Thesis Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Joseph Cohen Submitted to the Senate of the Technion
More informationAN EXTENSION TO THE BRUN-TITCHMARSH THEOREM
AN EXTENSION TO THE BRUN-TITCHMARSH THEOREM TSZ HO CHAN, STEPHEN KWOK-KWONG CHOI, AND KAI MAN TSANG Abstract The Siegel-Walfisz Theorem states that for any B > 0, we have p p a mod ϕ log for log B and,
More informationOn the counting function for the generalized Niven numbers
On the counting function for the generalized Niven numbers Ryan Daileda, Jessica Jou, Robert Lemke-Oliver, Elizabeth Rossolimo, Enrique Treviño Abstract Given an integer base q 2 and a completely q-additive
More informationArithmetic functions in short intervals
Arithmetic functions in short intervals Brad Rodgers Department of Mathematics University of Michigan Analytic Number Theory and Arithmetic Session JMM Jan. 2017 Brad Rodgers UM Short intervals Jan. 2017
More informationAdditive Combinatorics and Computational Complexity
Additive Combinatorics and Computational Complexity Luca Trevisan U.C. Berkeley Ongoing joint work with Omer Reingold, Madhur Tulsiani, Salil Vadhan Combinatorics: Studies: Graphs, hypergraphs, set systems
More informationPlaying Ball with the Largest Prime Factor
Playing Ball with the Largest Prime Factor An Introduction to Ruth-Aaron Numbers Madeleine Farris Wellesley College July 30, 2018 The Players The Players Figure: Babe Ruth Home Run Record: 714 The Players
More informationDense product-free sets of integers
Dense product-free sets of integers Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA Joint Mathematics Meetings Boston, January 6, 2012 Based on joint work with P. Kurlberg, J. C. Lagarias,
More informationModule 1: Analyzing the Efficiency of Algorithms
Module 1: Analyzing the Efficiency of Algorithms Dr. Natarajan Meghanathan Professor of Computer Science Jackson State University Jackson, MS 39217 E-mail: natarajan.meghanathan@jsums.edu What is an Algorithm?
More informationAppendix lecture 9: Extra terms.
Appendi lecture 9: Etra terms The Hyperolic method has een used to prove that d n = log + 2γ + O /2 n This can e used within the Convolution Method to prove Theorem There eists a constant C such that n
More informationON THE EQUATION τ(λ(n)) = ω(n) + k
ON THE EQUATION τ(λ(n)) = ω(n) + k A. GLIBICHUK, F. LUCA AND F. PAPPALARDI Abstract. We investigate some properties of the positive integers n that satisfy the equation τ(λ(n)) = ω(n)+k providing a complete
More informationOn prime factors of subset sums
On prime factors of subset sums by P. Erdös, A. Sárközy and C.L. Stewart * 1 Introduction For any set X let X denote its cardinality and for any integer n larger than one let ω(n) denote the number of
More informationAssignment 2 : Probabilistic Methods
Assignment 2 : Probabilistic Methods jverstra@math Question 1. Let d N, c R +, and let G n be an n-vertex d-regular graph. Suppose that vertices of G n are selected independently with probability p, where
More information1 Definition of the Riemann integral
MAT337H1, Introduction to Real Analysis: notes on Riemann integration 1 Definition of the Riemann integral Definition 1.1. Let [a, b] R be a closed interval. A partition P of [a, b] is a finite set of
More informationThe Probability that Random Polynomials Are Relatively r-prime
The Probability that Random Polynomials Are Relatively r-prime Kent E. Morrison Department of Mathematics California Polytechnic State University San Luis Obispo, CA 93407 kmorriso@calpoly.edu Zhou Dong
More informationSummatory function of the number of prime factors
Summatory function of the number of prime factor Xianchang Meng arxiv:80.06906v [math.nt] Jan 08 Abtract We conider the ummatory function of the number of prime factor for integer x over arithmetic progreion.
More informationA New Sieve for the Twin Primes
A ew Sieve for the Twin Primes and how the number of twin primes is related to the number of primes by H.L. Mitchell Department of mathematics CUY-The City College 160 Convent avenue ew York, Y 10031 USA
More informationThe ranges of some familiar arithmetic functions
The ranges of some familiar arithmetic functions Max-Planck-Institut für Mathematik 2 November, 2016 Carl Pomerance, Dartmouth College Let us introduce our cast of characters: ϕ, λ, σ, s Euler s function:
More informationSignals and Spectra - Review
Signals and Spectra - Review SIGNALS DETERMINISTIC No uncertainty w.r.t. the value of a signal at any time Modeled by mathematical epressions RANDOM some degree of uncertainty before the signal occurs
More informationInteger Sequences Avoiding Prime Pairwise Sums
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 11 (008), Article 08.5.6 Integer Sequences Avoiding Prime Pairwise Sums Yong-Gao Chen 1 Department of Mathematics Nanjing Normal University Nanjing 10097
More informationRoth s Theorem on Arithmetic Progressions
September 25, 2014 The Theorema of Szemerédi and Roth For Λ N the (upper asymptotic) density of Λ is the number σ(λ) := lim sup N Λ [1, N] N [0, 1] The Theorema of Szemerédi and Roth For Λ N the (upper
More informationSums and products. Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA. Dartmouth Mathematics Society May 16, 2012
Sums and products Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA Dartmouth Mathematics Society May 16, 2012 Based on joint work with P. Kurlberg, J. C. Lagarias, & A. Schinzel Let s begin
More informationThe Rainbow Turán Problem for Even Cycles
The Rainbow Turán Problem for Even Cycles Shagnik Das University of California, Los Angeles Aug 20, 2012 Joint work with Choongbum Lee and Benny Sudakov Plan 1 Historical Background Turán Problems Colouring
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 informationExact formulae for the prime counting function
Notes on Number Theory and Discrete Mathematics Vol. 19, 013, No. 4, 77 85 Exact formulae for the prime counting function Mladen Vassilev Missana 5 V. Hugo Str, 114 Sofia, Bulgaria e-mail: missana@abv.bg
More informationSOME REMARKS ON NUMBER THEORY BY P. ERDŐS 1. Let ABSTRACT This note contains some disconnected minor remarks on number theory. (1) Iz j I=1, 1<j<co be
SOME REMARKS ON NUMBER THEORY BY P. ERDŐS 1. Let ABSTRACT This note contains some disconnected minor remarks on number theory. (1) Iz j I=1, 1
More informationarxiv: v2 [math.nt] 15 Feb 2014
ON POLIGNAC S CONJECTURE FRED B. HOLT AND HELGI RUDD arxiv:1402.1970v2 [math.nt] 15 Feb 2014 Abstract. A few years ago we identified a recursion that works directly with the gaps among the generators in
More informationPRIME DIVISORS ARE POISSON DISTRIBUTED
February, 200 6: WSPC/INSTRUCTION FILE 000 International Journal of Number Theory Vol., No. (200) 8 c World Scientific Publishing Company PRIME DIVISORS ARE POISSON DISTRIBUTED ANDREW GRANVILLE Départment
More informationOn the counting function for the generalized. Niven numbers
On the counting function for the generalized Niven numbers R. C. Daileda, Jessica Jou, Robert Lemke-Oliver, Elizabeth Rossolimo, Enrique Treviño Introduction Let q 2 be a fied integer and let f be an arbitrary
More informationMath 680 Fall Chebyshev s Estimates. Here we will prove Chebyshev s estimates for the prime counting function π(x). These estimates are
Math 680 Fall 07 Chebyshev s Estimates Here we will prove Chebyshev s estimates for the prime coutig fuctio. These estimates are superseded by the Prime Number Theorem, of course, but are iterestig from
More informationLecture 2: Convergence of Random Variables
Lecture 2: Convergence of Random Variables Hyang-Won Lee Dept. of Internet & Multimedia Eng. Konkuk University Lecture 2 Introduction to Stochastic Processes, Fall 2013 1 / 9 Convergence of Random Variables
More informationLecture 2: CDF and EDF
STAT 425: Introduction to Nonparametric Statistics Winter 2018 Instructor: Yen-Chi Chen Lecture 2: CDF and EDF 2.1 CDF: Cumulative Distribution Function For a random variable X, its CDF F () contains all
More informationGOLDBACH S PROBLEMS ALEX RICE
GOLDBACH S PROBLEMS ALEX RICE Abstract. These are notes from the UGA Analysis and Arithmetic Combinatorics Learning Seminar from Fall 9, organized by John Doyle, eil Lyall, and Alex Rice. In these notes,
More informationSmol Results on the Möbius Function
Karen Ge August 3, 207 Introduction We will address how Möbius function relates to other arithmetic functions, multiplicative number theory, the primitive complex roots of unity, and the Riemann zeta function.
More informationRoth s Theorem on 3-term Arithmetic Progressions
Roth s Theorem on 3-term Arithmetic Progressions Mustazee Rahman 1 Introduction This article is a discussion about the proof of a classical theorem of Roth s regarding the existence of three term arithmetic
More informationFor more, see my paper of the same title!
Wild Partitions and Number Theory David P. Roberts University of Minnesota, Morris 1. Wild Partitions 2. Analytic Number Theory 3. Local algebraic number theory 4. Global algebraic number theory For more,
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 informationA CARLITZ MODULE ANALOGUE OF A CONJECTURE OF ERDŐS AND POMERANCE
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Number 9, September 009, Pages 459 459 S 000-994709)047-0 Article electronically published on April 6, 009 A CARLITZ MODULE ANALOGUE OF A CONJECTURE
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 informationThe Average Number of Divisors of the Euler Function
The Average Number of Divisors of the Euler Function Sungjin Kim Santa Monica College/Concordia University Irvine Department of Mathematics i707107@math.ucla.edu Dec 17, 2016 Euler Function, Carmichael
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 informationApplications of model theory in extremal graph combinatorics
Applications of model theory in extremal graph combinatorics Artem Chernikov (IMJ-PRG, UCLA) Logic Colloquium Helsinki, August 4, 2015 Szemerédi regularity lemma Theorem [E. Szemerédi, 1975] Every large
More informationThe ranges of some familiar arithmetic functions
The ranges of some familiar arithmetic functions Carl Pomerance Dartmouth College, emeritus University of Georgia, emeritus based on joint work with K. Ford, F. Luca, and P. Pollack and T. Freiburg, N.
More informationBig O Notation. P. Danziger
1 Comparing Algorithms We have seen that in many cases we would like to compare two algorithms. Generally, the efficiency of an algorithm can be guaged by how long it takes to run as a function of the
More informationBIASES IN PRIME FACTORIZATIONS AND LIOUVILLE FUNCTIONS FOR ARITHMETIC PROGRESSIONS
BIASES IN PRIME FACTORIZATIONS AND LIOUVILLE FUNCTIONS FOR ARITHMETIC PROGRESSIONS PETER HUMPHRIES, SNEHAL M. SHEKATKAR, AND TIAN AN WONG Abstract. We introduce a refinement of the classical Liouville
More informationArithmetic Randonnée An introduction to probabilistic number theory
Arithmetic Randonnée An introduction to probabilistic number theory E. Kowalski Version of June 4, 208 kowalski@math.ethz.ch Les probabilités et la théorie analytique des nombres, c est la même chose,
More informationThe dichotomy between structure and randomness. International Congress of Mathematicians, Aug Terence Tao (UCLA)
The dichotomy between structure and randomness International Congress of Mathematicians, Aug 23 2006 Terence Tao (UCLA) 1 A basic problem that occurs in many areas of analysis, combinatorics, PDE, and
More informationE-SYMMETRIC NUMBERS (PUBLISHED: COLLOQ. MATH., 103(2005), NO. 1, )
E-SYMMETRIC UMBERS PUBLISHED: COLLOQ. MATH., 032005), O., 7 25.) GAG YU Abstract A positive integer n is called E-symmetric if there exists a positive integer m such that m n = φm), φn)). n is called E-asymmetric
More informationON THE DIVISOR FUNCTION IN SHORT INTERVALS
ON THE DIVISOR FUNCTION IN SHORT INTERVALS Danilo Bazzanella Dipartimento di Matematica, Politecnico di Torino, Italy danilo.bazzanella@polito.it Autor s version Published in Arch. Math. (Basel) 97 (2011),
More information