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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 pn be the nth prime. Theorem (Goldston, Pintz, Yıldırım) lim inf

n→∞

pn+1 − pn log pn = 0.

Introduction Main Result Sketch of the Proof Applications

Bounded gaps between primes

Let pn be the nth prime. Theorem (Goldston, Pintz, Yıldırım) lim inf

n→∞

pn+1 − pn log pn = 0. Theorem (Zhang) lim inf

n→∞ (pn+1 − pn) ≤ C.

Introduction Main Result Sketch of the Proof Applications

Bounded gaps between primes

Let pn be the nth prime. Theorem (Goldston, Pintz, Yıldırım) lim inf

n→∞

pn+1 − pn log pn = 0. Theorem (Zhang) lim inf

n→∞ (pn+1 − pn) ≤ C.

Zhang: C ≤ 7 × 107,

Introduction Main Result Sketch of the Proof Applications

Bounded gaps between primes

Let pn be the nth prime. Theorem (Goldston, Pintz, Yıldırım) lim inf

n→∞

pn+1 − pn log pn = 0. Theorem (Zhang) lim inf

n→∞ (pn+1 − pn) ≤ C.

Zhang: C ≤ 7 × 107, Maynard-Tao: C ≤ 600,

Introduction Main Result Sketch of the Proof Applications

Bounded gaps between primes

Let pn be the nth prime. Theorem (Goldston, Pintz, Yıldırım) lim inf

n→∞

pn+1 − pn log pn = 0. Theorem (Zhang) lim inf

n→∞ (pn+1 − pn) ≤ C.

Zhang: C ≤ 7 × 107, Maynard-Tao: C ≤ 600, Polymath: C ≤ 270.

Introduction Main Result Sketch of the Proof Applications

Generalizations

Generalizations: Bounded gaps between Er 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, qn be the nth Er 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→∞ (qn+1 − qn) ≤ 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, qn be the nth 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→∞ (qn+1 − qn) ≤ 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, qn be the nth 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→∞ (qn+1 − qn) ≤ 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, qn be the nth 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→∞ (qn+1 − qn) ≤ 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. EP 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. EP number: a square-free number with all prime factors in P. Goal: Prove bounded gaps between EP numbers.

Introduction Main Result Sketch of the Proof Applications

M-admissible k-tuples

An M-admissible k-tuple is a set of linear forms {a1n + b1, . . . , akn + bk}

Introduction Main Result Sketch of the Proof Applications

M-admissible k-tuples

An M-admissible k-tuple is a set of linear forms {a1n + b1, . . . , akn + bk} 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 {a1n + b1, . . . , akn + bk} that never simultaneously represents all residue classes modulo p, for any prime p. satisfies M|ai and (M, ai/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 {a1n + b1, . . . , akn + bk} that never simultaneously represents all residue classes modulo p, for any prime p. satisfies M|ai and (M, ai/M) = 1 for each i. For our purpose, we often take a1 = · · · = ak = M.

Introduction Main Result Sketch of the Proof Applications

M-admissible k-tuples

An M-admissible k-tuple is a set of linear forms {a1n + b1, . . . , akn + bk} that never simultaneously represents all residue classes modulo p, for any prime p. satisfies M|ai and (M, ai/M) = 1 for each i. For our purpose, we often take a1 = · · · = ak = M. Density δi = the proportion of EP numbers represented by ain + bi.

Introduction Main Result Sketch of the Proof Applications

M-admissible k-tuples

An M-admissible k-tuple is a set of linear forms {a1n + b1, . . . , akn + bk} that never simultaneously represents all residue classes modulo p, for any prime p. satisfies M|ai and (M, ai/M) = 1 for each i. For our purpose, we often take a1 = · · · = ak = M. Density δi = the proportion of EP numbers represented by ain + bi. Minimum density δ = minimum of δi.

Introduction Main Result Sketch of the Proof Applications

M-admissible k-tuples

An M-admissible k-tuple is a set of linear forms {a1n + b1, . . . , akn + bk} that never simultaneously represents all residue classes modulo p, for any prime p. satisfies M|ai and (M, ai/M) = 1 for each i. For our purpose, we often take a1 = · · · = ak = M. Density δi = the proportion of EP numbers represented by ain + bi. Minimum density δ = minimum of δi. Strategy: If at least two Mn + bi represent EP numbers infinitely

• ften, our k-tuple gives bounded gaps (bounded by bk − b1).

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), {ain +bi} 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), {ain +bi} 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 EP numbers, provided that k > ν 41−α δφ(M) b(0, α)b(1, α) b(k, α) (1 − α)2Γ(α)Γ(1 − α), where b(k, α) := Γ(1 − α)Γ(k(1 − α) + 1) Γ((k + 1)(1 − α) + 1) .

Introduction Main Result Sketch of the Proof Applications

Precise Statement (twin-prime type)

As a corollary (for ν = 1, k = 2), we have a twin-prime type result.

Introduction Main Result Sketch of the Proof Applications

Precise Statement (twin-prime type)

As a corollary (for ν = 1, k = 2), we have a twin-prime type result. Let P be a set of prime with density α > 0 that satisfies SW (M), δ be the minimum density of the pair (Mn, Mn + d). Corollary For any even number d, assume that δφ(M) > F(α), where F(α) = 21−2α b(0, α)b(1, α) b(2, α) (1 − α)2Γ(α)Γ(1 − α). Then there are infinitely many n for which n and n + d are simultaneously EP numbers.

Introduction Main Result Sketch of the Proof Applications

Plot

Corollary For any even number d, assume that δφ(M) > F(α), then there are infinitely many n for which n and n + d are simultaneously EP numbers.

0.2 0.4 0.6 0.8 1.0 Α 5 10 15 20 25 FΑ

Introduction Main Result Sketch of the Proof Applications

Sketch of the proof

Following the same idea as GPY/GGPY and Thorne, consider S =

2N

• n=N

k

• i=1

χP(ain + bi) − ν   

• d|

i(ain+bi)

λd   

2

, where χP is the characteristic function of EP numbers, λd is any real numbers.

Introduction Main Result Sketch of the Proof Applications

Sketch of the proof

Following the same idea as GPY/GGPY and Thorne, consider S =

2N

• n=N

k

• i=1

χP(ain + bi) − ν   

• d|

i(ain+bi)

λd   

2

, where χP is the characteristic function of EP numbers, λd is any real numbers. If S > 0 for all large N, the theorem is true.

Introduction Main Result Sketch of the Proof Applications

Sketch of the proof

Following the same idea as GPY/GGPY and Thorne, consider S =

2N

• n=N

k

• i=1

χP(ain + bi) − ν

• 

   

• d|

i(ain+bi)

p∤d∀p∈P

λd     

2

, where χP is the characteristic function of EP numbers, λd is any real numbers. If S > 0 for all large N, the theorem is true. Remark: The additional requirement that p ∤ d for all p ∈ P gives a bigger sieve weight to those n for which

i(ain + bi) is divisible

by many primes in P.

Introduction Main Result Sketch of the Proof Applications

Sketch of the proof

Break S up into S0 :=

2N

• n=N

    

• d|

i(ain+bi)

p∤d∀p∈P

λd     

2

, and S1,j =

2N

• n=N

χP(ajn + bj)     

• d|

i(ain+bi)

p∤d∀p∈P

λd     

2

.

Introduction Main Result Sketch of the Proof Applications

Sketch of the proof

Break S up into S0 :=

2N

• n=N

    

• d|

i(ain+bi)

p∤d∀p∈P

λd     

2

, and S1,j =

2N

• n=N

χP(ajn + bj)     

• d|

i(ain+bi)

p∤d∀p∈P

λd     

2

. Then S = k

j=1 S1,j − νS0.

Introduction Main Result Sketch of the Proof Applications

Sketch of the proof

Break S up into S0 :=

2N

• n=N

    

• d|

i(ain+bi)

p∤d∀p∈P

λd     

2

, and S1,j =

2N

• n=N

χP(ajn + bj)     

• d|

i(ain+bi)

p∤d∀p∈P

λd     

2

. Then S = k

j=1 S1,j − νS0.

Choose the sieve weights λd so that S0 is small and S1,j is big.

Introduction Main Result Sketch of the Proof Applications

Sketch of the proof

Break S up into S0 :=

2N

• n=N

    

• d|

i(ain+bi)

p∤d∀p∈P

λd     

2

, and S1,j =

2N

• n=N

χP(ajn + bj)     

• d|

i(ain+bi)

p∤d∀p∈P

λd     

2

. Then S = k

j=1 S1,j − νS0.

Choose the sieve weights λd so that S0 is small and S1,j is big. Estimate S1,j, S0: use BV estimate and Selberg diagonalization.

Introduction Main Result Sketch of the Proof Applications

Applications

We now revisit two of Thorne’s applications: 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

Ideal class groups with order 4 elements

Thorne’s main result leads to the corollary: Corollary There are infinitely many pairs of E2 numbers m and n such that the class groups Cl(Q(√−m)) and Cl(Q(√−n)) each have elements of order 4, with |m − n| ≤ 64.

Introduction Main Result Sketch of the Proof Applications

Ideal class groups with order 4 elements

Our main result yields: Corollary There are infinitely many pairs of square-free m and n such that the class groups Cl(Q(√−m)) and Cl(Q(√−n)) each have elements of order 4, with |m − n| ≤ 8.

Introduction Main Result Sketch of the Proof Applications

Ideal class groups with order 4 elements

Theorem (Soundararajan) For any positive square-free number d ≡ 1 mod 8 whose prime factors ≡ ±1 mod 8, the class group Cl(Q( √ −d)) contains an element of order 4.

Introduction Main Result Sketch of the Proof Applications

Ideal class groups with order 4 elements

Theorem (Soundararajan) For any positive square-free number d ≡ 1 mod 8 whose prime factors ≡ ±1 mod 8, the class group Cl(Q( √ −d)) contains an element of order 4. P := the set of primes ≡ ±1 mod 8,

Introduction Main Result Sketch of the Proof Applications

Ideal class groups with order 4 elements

Theorem (Soundararajan) For any positive square-free number d ≡ 1 mod 8 whose prime factors ≡ ±1 mod 8, the class group Cl(Q( √ −d)) contains an element of order 4. P := the set of primes ≡ ±1 mod 8, δ = 1/2, α = 1/2, M = 8.

Introduction Main Result Sketch of the Proof Applications

Ideal class groups with order 4 elements

Theorem (Soundararajan) For any positive square-free number d ≡ 1 mod 8 whose prime factors ≡ ±1 mod 8, the class group Cl(Q( √ −d)) contains an element of order 4. P := the set of primes ≡ ±1 mod 8, δ = 1/2, α = 1/2, M = 8. Check that δφ(M) > F(α), therefore

Introduction Main Result Sketch of the Proof Applications

Ideal class groups with order 4 elements

Theorem (Soundararajan) For any positive square-free number d ≡ 1 mod 8 whose prime factors ≡ ±1 mod 8, the class group Cl(Q( √ −d)) contains an element of order 4. P := the set of primes ≡ ±1 mod 8, δ = 1/2, α = 1/2, M = 8. Check that δφ(M) > F(α), therefore Corollary There are infinitely many square-free numbers n such that the class groups Cl(Q(√−n)) and Cl(Q(√−n − 8)) each contain elements

• f order 4.

Introduction Main Result Sketch of the Proof Applications

Quadratic twists of Elliptic curves

Thorne’s main result leads to the corollary: Corollary Let E/Q be an elliptic curve without a Q-rational torsion point of

• rder 2. There is CE > 0 and infinitely many pairs of square-free

integers m and n for which: (i) L(E(m), 1) · L(E(n), 1) = 0, (ii) rank(E(m)) = rank(E(n)) = 0, (iii) |m − n| ≤ CE.

Introduction Main Result Sketch of the Proof Applications

Quadratic twists of Elliptic curves

Thorne’s main result leads to the corollary: Corollary Let E/Q be an elliptic curve without a Q-rational torsion point of

• rder 2. There is CE > 0 and infinitely many pairs of square-free

integers m and n for which: (i) L(E(m), 1) · L(E(n), 1) = 0, (ii) rank(E(m)) = rank(E(n)) = 0, (iii) |m − n| ≤ CE. For E := X0(11), Thorne calculated that CE ≤ 6152146.

Introduction Main Result Sketch of the Proof Applications

Quadratic twists of Elliptic curves

Thorne’s main result leads to the corollary: Corollary Let E/Q be an elliptic curve without a Q-rational torsion point of

• rder 2. There is CE > 0 and infinitely many pairs of square-free

integers m and n for which: (i) L(E(m), 1) · L(E(n), 1) = 0, (ii) rank(E(m)) = rank(E(n)) = 0, (iii) |m − n| ≤ CE. For E := X0(11), Thorne calculated that CE ≤ 6152146. Our result yields that CE ≤ 48.

Introduction Main Result Sketch of the Proof Applications

Quadratic twists of Elliptic curves

Thorne’s main result leads to the corollary: Corollary Let E/Q be an elliptic curve without a Q-rational torsion point of

• rder 2. There is CE > 0 and infinitely many pairs of square-free

integers m and n for which: (i) L(E(m), 1) · L(E(n), 1) = 0, (ii) rank(E(m)) = rank(E(n)) = 0, (iii) |m − n| ≤ CE. For E := X0(11), Thorne calculated that CE ≤ 6152146. Our result yields that CE ≤ 48. The proof uses results by Ono and Murty-Murty.

Introduction Main Result Sketch of the Proof Applications

Further improvement

Recall the main result: Corollary For any even number d, assume that δφ(M) > F(α), where F(α) = 21−2α b(0, α)b(1, α) b(2, α) (1 − α)2Γ(α)Γ(1 − α). Then there are infinitely many n for which n and n + d are simultaneously EP numbers.

Introduction Main Result Sketch of the Proof Applications

Further improvement

Recall the main result: Corollary For any even number d, assume that δφ(M) > F(α), where F(α) = 21−2α b(0, α)b(1, α) b(2, α) (1 − α)2Γ(α)Γ(1 − α). Then there are infinitely many n for which n and n + d are simultaneously EP numbers.

• J. Thorner has recently applied Maynard’s method to obtain

stronger results in these applications (if we consider only primes with the special properties).

Introduction Main Result Sketch of the Proof Applications

Further improvement

Recall the main result: Corollary For any even number d, assume that δφ(M) > F(α), where F(α) = 21−2α b(0, α)b(1, α) b(2, α) (1 − α)2Γ(α)Γ(1 − α). Then there are infinitely many n for which n and n + d are simultaneously EP numbers.

• J. Thorner has recently applied Maynard’s method to obtain

stronger results in these applications (if we consider only primes with the special properties). Using Maynard’s new sieve method may improve our main result.