New Sifting Iterations (bringing the combinatorics back) - PowerPoint PPT Presentation

New Sifting Iterations (bringing the combinatorics back) Zarathustra Brady

Sieve theoretic notation ◮ If A is a set of integers and P is a set of primes, then we define S ( A , P ) = { a ∈ A | ∀ p ∈ P , p ∤ a } . If z is a real number and P is the set of primes less than z , we abbreviate this to S ( A , z ) = { a ∈ A | ∀ p < z , p ∤ a } .

Sieve theoretic notation ◮ If A is a set of integers and P is a set of primes, then we define S ( A , P ) = { a ∈ A | ∀ p ∈ P , p ∤ a } . If z is a real number and P is the set of primes less than z , we abbreviate this to S ( A , z ) = { a ∈ A | ∀ p < z , p ∤ a } . ◮ For every squarefree d , we let A d be the set of multiples of d in A : A d = { a ∈ A | d | a } .

Sieve theoretic notation ◮ If A is a set of integers and P is a set of primes, then we define S ( A , P ) = { a ∈ A | ∀ p ∈ P , p ∤ a } . If z is a real number and P is the set of primes less than z , we abbreviate this to S ( A , z ) = { a ∈ A | ∀ p < z , p ∤ a } . ◮ For every squarefree d , we let A d be the set of multiples of d in A : A d = { a ∈ A | d | a } . ◮ This notation may be abused in various ways.

The dimension of a sieve ◮ Our running assumption is that there is a real number κ , called the sifting dimension , together with a multiplicative function, also called κ by abuse of notation, satisfying 0 ≤ κ ( p ) < p for all p and κ ( p )log( p ) � = ( κ + o (1)) log( x ) , p p ≤ x and that z , y are such that for every squarefree integer d , all of whose prime factors are less than z , we have � | A d | − κ ( d ) y � � � ≤ κ ( d ) . � � d

The dimension of a sieve ◮ Our running assumption is that there is a real number κ , called the sifting dimension , together with a multiplicative function, also called κ by abuse of notation, satisfying 0 ≤ κ ( p ) < p for all p and κ ( p )log( p ) � = ( κ + o (1)) log( x ) , p p ≤ x and that z , y are such that for every squarefree integer d , all of whose prime factors are less than z , we have � | A d | − κ ( d ) y � � � ≤ κ ( d ) . � � d ◮ This assumption may be weakened to � | A d | − κ ( d ) y y � � � ≤ κ ( d ) � � d log( y / d ) 2 κ + ǫ d without affecting the quality of sieve-theoretic bounds.

The dimension of a sieve: examples ◮ If A is an interval of length y , then we can take κ = 1, and for any d we will have � | A d | − y � � � ≤ 1 . � � d So searching for primes in an interval corresponds to a sieve of dimension 1.

The dimension of a sieve: examples ◮ If A is an interval of length y , then we can take κ = 1, and for any d we will have � | A d | − y � � � ≤ 1 . � � d So searching for primes in an interval corresponds to a sieve of dimension 1. ◮ If A = { n ( n + 2) | n ∈ [ x , x + y ) } , then |S ( A , √ x + y ) | counts the number of twin primes in the interval [ x , x + y ). This is a sieve of dimension 2.

The dimension of a sieve: examples ◮ If A is an interval of length y , then we can take κ = 1, and for any d we will have � | A d | − y � � � ≤ 1 . � � d So searching for primes in an interval corresponds to a sieve of dimension 1. ◮ If A = { n ( n + 2) | n ∈ [ x , x + y ) } , then |S ( A , √ x + y ) | counts the number of twin primes in the interval [ x , x + y ). This is a sieve of dimension 2. ◮ Counting numbers which can be written as a sum of two squares corresponds to a sieve with κ = 1 2 .

Fundamental Lemma of sieve theory ◮ The na¨ ıve approximation, using the Principle of Inclusion and Exclusion: � S ( A , z ) = µ ( d ) | A d | d | � p < z p µ ( d ) κ ( d ) y � ≈ d d | � p < z p � 1 − κ ( p ) � � = y . p p < z

Fundamental Lemma of sieve theory ◮ The na¨ ıve approximation, using the Principle of Inclusion and Exclusion: � S ( A , z ) = µ ( d ) | A d | d | � p < z p µ ( d ) κ ( d ) y � ≈ d d | � p < z p � 1 − κ ( p ) � � = y . p p < z ◮ If y = z s with s fixed, this is within a constant factor of the truth!

Fundamental Lemma of sieve theory Lemma (Selberg) Define functions f κ ( s ) , F κ ( s ) with f κ ( s ) as large as possible and F κ ( s ) as small as possible such that if y = z s with s fixed and z going to infinity, then S ( A , z ) � ≤ F κ ( s ) + o (1) f κ ( s ) + o (1) ≤ � 1 − κ ( p ) y � p < z p for any weighted set A satisfying our basic assumption. Then the functions f κ ( s ) , F κ ( s ) are finite, continuous, monotone, and computable for s > 1 , and they tend to 1 exponentially as s goes to infinity.

What are the sifting functions f κ , F κ ? ◮ The precise values of f κ , F κ are only known in two cases: κ = 1 2 and κ = 1.

What are the sifting functions f κ , F κ ? ◮ The precise values of f κ , F κ are only known in two cases: κ = 1 2 and κ = 1. ◮ When κ = 1, writing f = f 1 and F = F 1 , we have F ( s ) = 2 e γ 1 ≤ s ≤ 3 s d ds ( sF ( s )) = f ( s − 1) s ≥ 3 f ( s ) = 2 e γ log( s − 1) 2 ≤ s ≤ 4 s d ds ( sf ( s )) = F ( s − 1) s ≥ 2

Sifting Limit ◮ Often we are interested in proving a nontrivial lower bound on the size of the set S ( A , z ) (for instance, we would like to prove that twin primes exist). In other words, we want to show that f κ ( s ) > 0.

Sifting Limit ◮ Often we are interested in proving a nontrivial lower bound on the size of the set S ( A , z ) (for instance, we would like to prove that twin primes exist). In other words, we want to show that f κ ( s ) > 0. ◮ We define the sifting limit , β κ , to be β κ = inf { s | f κ ( s ) > 0 } . If β κ < 2, then we win!

Sifting Limit ◮ Often we are interested in proving a nontrivial lower bound on the size of the set S ( A , z ) (for instance, we would like to prove that twin primes exist). In other words, we want to show that f κ ( s ) > 0. ◮ We define the sifting limit , β κ , to be β κ = inf { s | f κ ( s ) > 0 } . If β κ < 2, then we win! 2 = 1, β 1 = 2. For 1 ◮ β 1 2 < κ < 1, we have β κ < 2 κ .

Sifting Limit ◮ Often we are interested in proving a nontrivial lower bound on the size of the set S ( A , z ) (for instance, we would like to prove that twin primes exist). In other words, we want to show that f κ ( s ) > 0. ◮ We define the sifting limit , β κ , to be β κ = inf { s | f κ ( s ) > 0 } . If β κ < 2, then we win! 2 = 1, β 1 = 2. For 1 ◮ β 1 2 < κ < 1, we have β κ < 2 κ . ◮ Selberg: if κ is sufficiently large, then β < 2 κ + 0 . 4454.

Sifting Limit ◮ Often we are interested in proving a nontrivial lower bound on the size of the set S ( A , z ) (for instance, we would like to prove that twin primes exist). In other words, we want to show that f κ ( s ) > 0. ◮ We define the sifting limit , β κ , to be β κ = inf { s | f κ ( s ) > 0 } . If β κ < 2, then we win! 2 = 1, β 1 = 2. For 1 ◮ β 1 2 < κ < 1, we have β κ < 2 κ . ◮ Selberg: if κ is sufficiently large, then β < 2 κ + 0 . 4454. ◮ Diamond-Halberstam-Richert: β 3 2 ≤ 3 . 11582 ... , β 2 ≤ 4 . 26645 ... .

Buchstab iteration ◮ When κ ≤ 1, the best known sieves are based on Buchstab’s identity: � S ( A , z ) = | A | − S ( A p , p ) . p < z

Buchstab iteration ◮ When κ ≤ 1, the best known sieves are based on Buchstab’s identity: � S ( A , z ) = | A | − S ( A p , p ) . p < z ◮ This leads to the inequalities � s κ f κ ( s ) ≥ s κ − κ t κ − 1 ( F κ ( t − 1) − 1) dt , t > s � s κ F κ ( s ) ≤ s κ + κ t κ − 1 (1 − f κ ( t − 1)) dt . t > s

Buchstab iteration ◮ When κ ≤ 1, the best known sieves are based on Buchstab’s identity: � S ( A , z ) = | A | − S ( A p , p ) . p < z ◮ This leads to the inequalities � s κ f κ ( s ) ≥ s κ − κ t κ − 1 ( F κ ( t − 1) − 1) dt , t > s � s κ F κ ( s ) ≤ s κ + κ t κ − 1 (1 − f κ ( t − 1)) dt . t > s ◮ Iterative application of these inequalities leads to the β -sieve.

Buchstab iteration ◮ When κ ≤ 1, the best known sieves are based on Buchstab’s identity: � S ( A , z ) = | A | − S ( A p , p ) . p < z ◮ This leads to the inequalities � s κ f κ ( s ) ≥ s κ − κ t κ − 1 ( F κ ( t − 1) − 1) dt , t > s � s κ F κ ( s ) ≤ s κ + κ t κ − 1 (1 − f κ ( t − 1)) dt . t > s ◮ Iterative application of these inequalities leads to the β -sieve. ◮ When κ is 1 2 or 1, we have equality!

Equality case: the parity problem ◮ Define weighted sets A + , A − , supported on [1 , y ], so that the weight A + assigns to n is 1 − λ ( n ) and the weight A − assigns to n is 1 + λ ( n ).

Equality case: the parity problem ◮ Define weighted sets A + , A − , supported on [1 , y ], so that the weight A + assigns to n is 1 − λ ( n ) and the weight A − assigns to n is 1 + λ ( n ). ◮ These weighted sets satisfy Buchstab-like identities: for any w ≤ z , we have � S ( A + , z ) = S ( A + , w ) − S ( A − p , p ) w < p < z and � S ( A − , z ) = S ( A − , w ) − S ( A + p , p ) . w < p < z