Bounded gaps between products of special primes Ping Ngai (Brian) - PowerPoint PPT Presentation

Introduction Main Result Sketch of the Proof Applications Bounded gaps between products of special primes Ping Ngai (Brian) Chung (MIT), Shiyu Li (UC Berkeley) January 17, 2014

Introduction Main Result Sketch of the Proof Applications Bounded gaps between primes Let p n be the n th prime. Theorem (Goldston, Pintz, Yıldırım) p n +1 − p n lim inf = 0 . log p n n →∞

Introduction Main Result Sketch of the Proof Applications Bounded gaps between primes Let p n be the n th prime. Theorem (Goldston, Pintz, Yıldırım) p n +1 − p n lim inf = 0 . log p n n →∞ Theorem (Zhang) lim inf n →∞ ( p n +1 − p n ) ≤ C .

Introduction Main Result Sketch of the Proof Applications Bounded gaps between primes Let p n be the n th prime. Theorem (Goldston, Pintz, Yıldırım) p n +1 − p n lim inf = 0 . log p n n →∞ Theorem (Zhang) lim inf n →∞ ( p n +1 − p n ) ≤ C . Zhang: C ≤ 7 × 10 7 ,

Introduction Main Result Sketch of the Proof Applications Bounded gaps between primes Let p n be the n th prime. Theorem (Goldston, Pintz, Yıldırım) p n +1 − p n lim inf = 0 . log p n n →∞ Theorem (Zhang) lim inf n →∞ ( p n +1 − p n ) ≤ C . Zhang: C ≤ 7 × 10 7 , Maynard-Tao: C ≤ 600,

Introduction Main Result Sketch of the Proof Applications Bounded gaps between primes Let p n be the n th prime. Theorem (Goldston, Pintz, Yıldırım) p n +1 − p n lim inf = 0 . log p n n →∞ Theorem (Zhang) lim inf n →∞ ( p n +1 − p n ) ≤ C . Zhang: C ≤ 7 × 10 7 , Maynard-Tao: C ≤ 600, Polymath: C ≤ 270.

Introduction Main Result Sketch of the Proof Applications Generalizations Generalizations: Bounded gaps between E r numbers (product of r distinct primes) for r ≥ 2? Restrict to a “nice” subset of primes P ?

Introduction Main Result Sketch of the Proof Applications Work of Thorne Let P be a “well-distributed” subset of primes, q n be the n th E r number with all prime factors in P . Theorem (Thorne) For any such P and r ≥ 2 , there exists an explicit constant C ( r , P ) such that lim inf n →∞ ( q n +1 − q n ) ≤ C ( r , P ) .

Introduction Main Result Sketch of the Proof Applications Applications Thorne’s main result leads to several corollaries: Bounded gaps between integers n with special properties: 1 The class group Cl ( Q ( √− n )) contains order 4 elements. 2 Given an elliptic curve E / Q , the quadratic twist E ( n ) has rank 0 and L ( E ( n ) , 1) � = 0.

Introduction Main Result Sketch of the Proof Applications Our observation We observe that in Thorne’s applications, without a restriction on the number of prime factors, much improved bounds are possible.

Introduction Main Result Sketch of the Proof Applications Our observation We observe that in Thorne’s applications, without a restriction on the number of prime factors, much improved bounds are possible. Let P be a “well-distributed” subset of primes, q n be the n th square-free number with all prime factors in P . Theorem (C., Li) For any such P , there exists an explicit constant C ( P ) such that lim inf n →∞ ( q n +1 − q n ) ≤ C ( P ) .

Introduction Main Result Sketch of the Proof Applications Our observation We observe that in Thorne’s applications, without a restriction on the number of prime factors, much improved bounds are possible. Let P be a “well-distributed” subset of primes, q n be the n th square-free number with all prime factors in P . Theorem (C., Li) For any such P , there exists an explicit constant C ( P ) such that lim inf n →∞ ( q n +1 − q n ) ≤ C ( P ) . Here, the constant C ( P ) is smaller than C ( r , P ) for all r ≥ 2.

Introduction Main Result Sketch of the Proof Applications Our observation We observe that in Thorne’s applications, without a restriction on the number of prime factors, much improved bounds are possible. Let P be a “well-distributed” subset of primes, q n be the n th square-free number with all prime factors in P . Theorem (C., Li) For any such P , there exists an explicit constant C ( P ) such that lim inf n →∞ ( q n +1 − q n ) ≤ C ( P ) . Here, the constant C ( P ) is smaller than C ( r , P ) for all r ≥ 2. We will make the statement precise. Then we revisit Thorne’s examples and give better bounds on gaps.

Introduction Main Result Sketch of the Proof Applications Regularity condition on P Following Thorne, we require that P is a set of primes with positive “nice” density.

Introduction Main Result Sketch of the Proof Applications Regularity condition on P Following Thorne, we require that P is a set of primes with positive “nice” density. P is well-distributed in arithmetic progressions: P satisfies a Siegel-Walfisz condition SW ( M ),

Introduction Main Result Sketch of the Proof Applications Regularity condition on P Following Thorne, we require that P is a set of primes with positive “nice” density. P is well-distributed in arithmetic progressions: P satisfies a Siegel-Walfisz condition SW ( M ), where M is an exceptional modulus: we allow bad distribution modulo q for ( q , M ) > 1.

Introduction Main Result Sketch of the Proof Applications Regularity condition on P Following Thorne, we require that P is a set of primes with positive “nice” density. P is well-distributed in arithmetic progressions: P satisfies a Siegel-Walfisz condition SW ( M ), where M is an exceptional modulus: we allow bad distribution modulo q for ( q , M ) > 1. E P number: a square-free number with all prime factors in P .

Introduction Main Result Sketch of the Proof Applications Regularity condition on P Following Thorne, we require that P is a set of primes with positive “nice” density. P is well-distributed in arithmetic progressions: P satisfies a Siegel-Walfisz condition SW ( M ), where M is an exceptional modulus: we allow bad distribution modulo q for ( q , M ) > 1. E P number: a square-free number with all prime factors in P . Goal: Prove bounded gaps between E P numbers.

Introduction Main Result Sketch of the Proof Applications M -admissible k -tuples An M -admissible k -tuple is a set of linear forms { a 1 n + b 1 , . . . , a k n + b k }

Introduction Main Result Sketch of the Proof Applications M -admissible k -tuples An M -admissible k -tuple is a set of linear forms { a 1 n + b 1 , . . . , a k n + b k } that never simultaneously represents all residue classes modulo p , for any prime p .

Introduction Main Result Sketch of the Proof Applications M -admissible k -tuples An M -admissible k -tuple is a set of linear forms { a 1 n + b 1 , . . . , a k n + b k } that never simultaneously represents all residue classes modulo p , for any prime p . satisfies M | a i and ( M , a i / M ) = 1 for each i .

Introduction Main Result Sketch of the Proof Applications M -admissible k -tuples An M -admissible k -tuple is a set of linear forms { a 1 n + b 1 , . . . , a k n + b k } that never simultaneously represents all residue classes modulo p , for any prime p . satisfies M | a i and ( M , a i / M ) = 1 for each i . For our purpose, we often take a 1 = · · · = a k = M .

Introduction Main Result Sketch of the Proof Applications M -admissible k -tuples An M -admissible k -tuple is a set of linear forms { a 1 n + b 1 , . . . , a k n + b k } that never simultaneously represents all residue classes modulo p , for any prime p . satisfies M | a i and ( M , a i / M ) = 1 for each i . For our purpose, we often take a 1 = · · · = a k = M . Density δ i = the proportion of E P numbers represented by a i n + b i .

Introduction Main Result Sketch of the Proof Applications M -admissible k -tuples An M -admissible k -tuple is a set of linear forms { a 1 n + b 1 , . . . , a k n + b k } that never simultaneously represents all residue classes modulo p , for any prime p . satisfies M | a i and ( M , a i / M ) = 1 for each i . For our purpose, we often take a 1 = · · · = a k = M . Density δ i = the proportion of E P numbers represented by a i n + b i . Minimum density δ = minimum of δ i .

Introduction Main Result Sketch of the Proof Applications M -admissible k -tuples An M -admissible k -tuple is a set of linear forms { a 1 n + b 1 , . . . , a k n + b k } that never simultaneously represents all residue classes modulo p , for any prime p . satisfies M | a i and ( M , a i / M ) = 1 for each i . For our purpose, we often take a 1 = · · · = a k = M . Density δ i = the proportion of E P numbers represented by a i n + b i . Minimum density δ = minimum of δ i . Strategy: If at least two Mn + b i represent E P numbers infinitely often, our k -tuple gives bounded gaps (bounded by b k − b 1 ).

Introduction Main Result Sketch of the Proof Applications Precise Statement Let P be a set of prime with density α > 0 that satisfies SW ( M ), { a i n + b i } be an M -admissible k -tuple with minimum density δ .

Introduction Main Result Sketch of the Proof Applications Precise Statement Let P be a set of prime with density α > 0 that satisfies SW ( M ), { a i n + b i } be an M -admissible k -tuple with minimum density δ . Theorem (C., Li) There are at least ν + 1 forms among them which infinitely often simultaneously represent E P numbers, provided that k > ν 4 1 − α b (0 , α ) b (1 , α ) (1 − α ) 2 Γ( α )Γ(1 − α ) , δφ ( M ) b ( k , α ) where b ( k , α ) := Γ(1 − α )Γ( k (1 − α ) + 1) Γ(( k + 1)(1 − α ) + 1) .