The Density Hypothesis for Dirichlet L-Series

Vinogradov, A. I.

doi:10.1142/9789812776600_0019

Cited by 31 publications

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“…This reduces the problem to estimating the number of prime values of L j (n) for n ∈ [X, 2X] in many different arithmetic progressions with moduli of size about R 2 . The Elliott-Halberstam conjecture [18] asserts that we should be able to do this when R 2 < X 1−ǫ , but unconditionally we only know how to do this when R 2 < X 1/2−ǫ , using the Bombieri-Vinogradov Theorem [7,73]. After some computation one finds that, provided we do have suitable estimates for primes in arithmetic progressions, the choice (3.1) gives…”

Section: The Gpy Methods and Weak Prime K-tuplesmentioning

confidence: 99%

The twin prime conjecture

Maynard

2019

Jpn. J. Math.

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Section: The Gpy Methods and Weak Prime K-tuplesmentioning

confidence: 99%

The twin prime conjecture

Maynard

2019

Jpn. J. Math.

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“…This reduces the problem to estimating the number of prime values of L j (n) for n ∈ [X, 2X] in many different arithmetic progressions with moduli of size about R 2 . The Elliott-Halberstam conjecture [16] asserts that we should be able to do this when R 2 < X 1−ǫ , but unconditionally we only know how to do this when R 2 < X 1/2−ǫ , using the Bombieri-Vinogradov Theorem [8,62]. After some computation one finds that, provided we do have suitable estimates for primes in arithmetic progressions, the choice (2.1) gives…”

Section: The Gpy Sieve Methodsmentioning

confidence: 99%

Gaps Between Primes

Maynard

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…Vinogradov [50] on (weighted) equidistribution of primes in arithmetic progressions; see below. By assuming a strong hypothesis of Elliott and Halberstam [12] on this kind of equidistribution, Goldston, Pintz and Yildirim could actually prove that lim inf n→∞ ( p n+1 − p n ) ≤ 16; (2.3) cf.…”

Section: Distribution Of the Primesmentioning

confidence: 99%

Prime pairs and the zeta function

Korevaar

2009

Journal of Approximation Theory

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Are there infinitely many prime pairs with given even difference? Most mathematicians think so. Using a strong arithmetic hypothesis, Goldston, Pintz and Yildirim have recently shown that there are infinitely many pairs of primes differing by at most sixteen.There is extensive numerical support for the prime-pair conjecture (PPC) of Hardy and Littlewood [G.H. Hardy, J.E. Littlewood, Some problems of 'partitio numerorum'. III: On the expression of a number as a sum of primes, Acta Math. 44 (1923) 1-70 (sec. 3)] on the asymptotic behavior of π 2r (x), the number of prime pairs ( p, p + 2r ) with p ≤ x. Assuming Riemann's Hypothesis (RH), Montgomery and others have studied the pair-correlation of zeta's complex zeros, indicating connections with the PPC. Using a Tauberian approach, the author shows that the PPC is equivalent to specific boundary behavior of a function involving zeta's complex zeros. A certain hypothesis on equidistribution of prime pairs, or a speculative supplement to Montgomery's work on pair-correlation, would imply that there is an abundance of prime pairs.

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The Density Hypothesis for Dirichlet L-Series

Cited by 31 publications

References 0 publications

The twin prime conjecture

The twin prime conjecture

Gaps Between Primes

Prime pairs and the zeta function

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