Restriction theory of the Selberg sieve, with applications

Green, Ben; Tao, Terence

doi:10.5802/jtnb.538

Cited by 65 publications

(115 citation statements)

References 32 publications

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“…By (3.2), θ ∈ I (θ j , /8N ) for some j ∈ J . Hence The remaining arguments go along the same line as those of Green [9,11]. Define…”

Section: Decomposition Of a I Into Uniform And Anti-uniform Partsmentioning

confidence: 96%

“…Here, we have used the prime number theorem in arithmetic progressions of modulus W Y , which is again a constant. Finally, the discrete majorant property for a i follows from the result of Green and Tao [11]. where the singular series G F is defined by…”

Section: Proof Of Vinogradov's Theoremmentioning

confidence: 99%

“…(See (1.2) and (1.7) in [11]). In the current case, γ ( p) = 1 for p w and γ ( p) = 1 − 1/ p for p > w. Hence…”

Section: Proof Of Vinogradov's Theoremmentioning

confidence: 99%

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An -Function-Free Proof of Vinogradov’s Three Primes Theorem

Shao

2014

Forum math. Sigma

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We give a new proof of Vinogradov's three primes theorem, which asserts that all sufficiently large odd positive integers can be written as the sum of three primes. Existing proofs rely on the theory of L-functions, either explicitly or implicitly. Our proof is sieve theoretical and uses a transference principle, the idea of which was first developed by Green [Ann. of Math. (2008), . To make our argument work, we also develop an additive combinatorial result concerning popular sums, which may be of independent interest.

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“…By (3.2), θ ∈ I (θ j , /8N ) for some j ∈ J . Hence The remaining arguments go along the same line as those of Green [9,11]. Define…”

Section: Decomposition Of a I Into Uniform And Anti-uniform Partsmentioning

confidence: 96%

Section: Proof Of Vinogradov's Theoremmentioning

confidence: 99%

See 1 more Smart Citation

An -Function-Free Proof of Vinogradov’s Three Primes Theorem

Shao

2014

Forum math. Sigma

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“…This means that, according to [GT,Prop. 3.1], β : Z + → R is a non-negative function satisfying the majorant property…”

Section: ])mentioning

confidence: 99%

“…Applying [GT,Prop. 4.2] with F (n) = b + nM and R = N ′1/10 , we obtain that, for p > 2 and any complex sequence (b n ) n , (2.5)…”

Section: ])mentioning

confidence: 99%

Improving Roth's Theorem in the Primes

Helfgott

Roton

2010

International Mathematics Research Notices

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Abstract. Let A be a subset of the primes. Let δP (N ) = |{n ∈ A : n ≤ N }| |{n prime : n ≤ N }| . We prove that, if δP (N ) ≥ C log log log N (log log N ) 1/3 for N ≥ N0, where C and N0 are absolute constants, then A ∩ [1, N ] contains a non-trivial three-term arithmetic progression.This improves on Green's result [Gr], which needs δP (N ) ≥ C ′ s log log log log log N log log log log N .

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The Thirties

Narkiewicz

2012

Springer Monographs in Mathematics

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Restriction theory of the Selberg sieve, with applications

Cited by 65 publications

References 32 publications

An -Function-Free Proof of Vinogradov’s Three Primes Theorem

An -Function-Free Proof of Vinogradov’s Three Primes Theorem

Improving Roth's Theorem in the Primes

The Thirties

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