CTNT 2020: Introduction to Sieves Brandon Alberts University of - PowerPoint PPT Presentation

CTNT 2020: Introduction to Sieves Brandon Alberts University of Connecticut June 2020 Brandon Alberts CTNT 2020: Introduction to Sieves

Introduction Brandon Alberts CTNT 2020: Introduction to Sieves

Definition A sieve is a tool for separating desired objects from other objects. Examples: A pasta strainer separates pasta from water. Sieves were used during the gold rush to separate gold from sand and dirt. The Sieve of Eratosthenes separates primes numbers from all other numbers. Brandon Alberts CTNT 2020: Introduction to Sieves

Sieve of Eratosthenes 65 75 74 73 72 71 70 69 68 67 66 64 77 63 62 61 60 59 58 57 56 55 54 76 78 52 91 Brandon Alberts 100 99 98 97 96 95 94 93 92 90 79 89 88 87 86 85 84 83 82 81 80 53 51 1 13 23 22 21 20 19 18 17 16 15 14 12 25 11 10 9 8 7 6 5 4 3 2 24 26 50 39 49 48 47 46 45 44 43 42 41 40 38 27 37 36 35 34 33 32 31 30 29 28 CTNT 2020: Introduction to Sieves

Asymptotic Notation and Arithmetic Functions Brandon Alberts CTNT 2020: Introduction to Sieves

Definition An arithmetic function is a function f : N C . These functions can be used to capture and study certain arithmetic behaviors. Brandon Alberts CTNT 2020: Introduction to Sieves

Arithmetic functions often have very erratic behavior, which makes them more difficult to deal with using analytic techniques. Consider the divisor # positive divisors of n . (see CoCalc) Brandon Alberts CTNT 2020: Introduction to Sieves function d n

We can smooth out the information contained in an arithmetic function n x d n Brandon Alberts CTNT 2020: Introduction to Sieves by considering the function of a real variable x

Definition if Brandon Alberts 1 . g x f x x lim g x Let f x and write f x is . We say f x be two functions and let x and g x CTNT 2020: Introduction to Sieves asymptotic to g x

Lemma x x Brandon Alberts CTNT 2020: Introduction to Sieves

Definition f x C g x for all x x 0 . Equivalently, lim sup x g x Let f x . is “asymptotically the same order of magnitude or Exercise: If f x g x then f x O g x . Brandon Alberts 0 such that f x CTNT 2020: Introduction to Sieves g x , if or f x and g x be function of the real variable x . We define the following: (a) f x is little-oh of g x , written f x o g x , if lim x f x g x 0 . is “asymptotically smaller” than g x . (b) f x is big-oh of g x , written f x O g x In this case, f x there exists a constant C In this case, f x smaller” than g x .

Definition We write Brandon Alberts . belong to the same coset when quotienting by the and g x for x sufficiently large. The above notation is then equivalent to stating are ideals in the ring of functions defined and o h x CTNT 2020: Introduction to Sieves Similar notation applies to little-oh. . O h x g x f x to mean O h x g x f x Exercise: O h x that f x ideal O h x

Lemma x x main term O 1 error term Brandon Alberts CTNT 2020: Introduction to Sieves

Exercises: O , then f x O h x . 4 If f x O g x , then n n and g x . 5 If f x O g x and y is some real number, then x O x Brandon Alberts 1 O h x O g x If f x f x O g x O f x g x and f x o g x o f x g x , 2 CTNT 2020: Introduction to Sieves If f x O g x and h x O g x then f x h x O g x , 3 x f n x g n y f t dt y g t dt .

Abel Summation Brandon Alberts CTNT 2020: Introduction to Sieves

Abel Summation A t f t f t and “ dv ” a n : y n x a n f n x Theorem y x y A t f t dt Brandon Alberts A t is like a “discrete antiderivative” of a n , and we can recognize the t dt . A t f y Write A x n the interval y , x for y x . Then y n x a n f n A x f x A y f y x CTNT 2020: Introduction to Sieves x a n and suppose f t is a differentiable function on familar integration by parts formula with u

Corollary n x 1 n log x O 1 Brandon Alberts CTNT 2020: Introduction to Sieves

Theorem n x d n x log x O x Brandon Alberts CTNT 2020: Introduction to Sieves

M¨ obius function Brandon Alberts CTNT 2020: Introduction to Sieves

Definition Lemma Brandon Alberts 1 n 0 1 n 1 µ d d n otherwise The M¨ 0 1 i k n µ n obius function is defined as follows: CTNT 2020: Introduction to Sieves 1 k p i is a product of k distinct primes

Theorem (Mobius Inversion) If f n d n g d then g n d n µ d f n d . Brandon Alberts CTNT 2020: Introduction to Sieves

Squarefree numbers 65 75 74 73 72 71 70 69 68 67 66 64 77 63 62 61 60 59 58 57 56 55 54 76 78 52 91 Brandon Alberts 100 99 98 97 96 95 94 93 92 90 79 89 88 87 86 85 84 83 82 81 80 53 51 1 13 23 22 21 20 19 18 17 16 15 14 12 25 11 10 9 8 7 6 5 4 3 2 24 26 50 39 49 48 47 46 45 44 43 42 41 40 38 27 37 36 35 34 33 32 31 30 29 28 CTNT 2020: Introduction to Sieves

Theorem # squarefree numbers x n 1 µ n n 2 x O x Brandon Alberts CTNT 2020: Introduction to Sieves

Prime Numbers Brandon Alberts CTNT 2020: Introduction to Sieves

Theorem (Prime Number Theorem) Let π x # primes p x p π x x log x There are variety of proofs of the PNT, the most accessible of which require complex analytic techniques. There does exist an “elementary proof” (i.e. one that does not appeal to complex analysis), but it is too long to treat in this course. Brandon Alberts CTNT 2020: Introduction to Sieves x 1 . Then

Theorem (Chebysheff’s Theorem) π x O x log x O x log x if and only if θ x : p x O x . Brandon Alberts CTNT 2020: Introduction to Sieves Exercise: π x log p

Theorem p x log p p log x O 1 Brandon Alberts CTNT 2020: Introduction to Sieves

Corollary p x 1 p log log x O 1 Exercise: Prove this corollary using Abel Summation with f t log t 1 . Brandon Alberts CTNT 2020: Introduction to Sieves

Sieve of Eratosthenes Brandon Alberts CTNT 2020: Introduction to Sieves

Eratosthenes sieved out numbers divisible by small primes. We can this p Brandon Alberts O 2 z p 1 1 z x by considering the function Φ x , z Theorem where x and z are positive real numbers. z x : n is not divisible by any primes # n Φ x , z CTNT 2020: Introduction to Sieves

Ψ x , z # n x : if p n then p z Brandon Alberts CTNT 2020: Introduction to Sieves To improve on the error O 2 z , consider the function

Theorem Ψ x , z x log z exp log x log z Brandon Alberts CTNT 2020: Introduction to Sieves

Theorem Φ x , z x p z 1 1 p O log x log z Brandon Alberts CTNT 2020: Introduction to Sieves x log z 2 exp

Theorem (Merten’s Theorem) p z 1 1 p e γ Brandon Alberts CTNT 2020: Introduction to Sieves log z , where γ is the Euler-Mascheroni constant.

Let A be a set of integers to ω p distinct residue classes modulo p . Brandon Alberts p P z A # Define S A , P , z A be a subset of integers belonging x , P a set of primes, and P z P , let A p For each prime p p . z p p P CTNT 2020: Introduction to Sieves A p .

If d is a squarefree number divisible by primes of P , define ω d Set ω 1 1 and A 1 A . Brandon Alberts CTNT 2020: Introduction to Sieves p d ω p and A d p d A p .

Theorem (The sieve of Eratosthenes) W z O X y log z log y log z where p P Suppose the following conditions hold: p z 1 ω p p . Brandon Alberts XW z S A , P , z Then p P z There exists an X such that # A d ω d d X O ω d , For some κ 0 , ω p log p y . p κ log z O 1 , For some y 0 , # A d CTNT 2020: Introduction to Sieves 0 for every d log z κ 1 exp

Brun’s Sieves Brandon Alberts CTNT 2020: Introduction to Sieves

Brun’s sieve is set up in essentially the same way as Eratosthenes. Given we want to remove, and measure the size of S A , P , z # A p P z A p . Brandon Alberts CTNT 2020: Introduction to Sieves some set A of integers x , we have some collection of A p of elements

Idea (Punchline of Brun’s results) Under similar, but slightly relaxed, hypotheses to the sieve of Eratosthenes, Burn proves that S A , P , z XW z O better error # A and W z p P z 1 ω p p . Brandon Alberts CTNT 2020: Introduction to Sieves where X

Theorem (Brun’s Pure Sieve) Suppose the following conditions hold: Brandon Alberts where η is any positive number (possibly depending on x and z .) error terms log z XW z O main term XW z S A , P , z Then C 2 , CTNT 2020: Introduction to Sieves p ω p p P z C , There exists a constant C such that ω p , O ω d X d ω d There exists an X such that # A d There exist constants C 1 and C 2 such that C 1 log log z η log η O z η log log z

Theorem # p x : p and p 2 are prime . Brandon Alberts CTNT 2020: Introduction to Sieves x log log x 2 log x 2

Corollary p p 2 prime 1 p . Brandon Alberts CTNT 2020: Introduction to Sieves

The big idea: truncated M¨ ν d Brandon Alberts µ d 1 r ν n d n θ µ d r d n obius Inversion µ n d n that 1 such # distinct prime divisors of n . There exists θ ν n r Let n and r be positive integers with Lemma CTNT 2020: Introduction to Sieves