Oscillation theorems for primes in arithmetic progressions and for sifting functions

Friedlander, John B.; Granville, Andrew; Hildebrand, Adolf; Maier, Helmut

doi:10.1090/s0894-0347-1991-1080647-5

Cited by 28 publications

(27 citation statements)

References 12 publications

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“…A. Hildebrand and Maier [14] had previously shown such a result for y ≤ exp((log x) 1 3 −ε ) (more precisely they obtained a bound y −(1+o(1))τ /(1−τ ) in the range 0 < τ < 1/3), and were able to obtain our result assuming the validity of the Generalized Riemann Hypothesis. We have also been able to extend the uniformity with which Friedlander and Granville's result (1.1) holds, obtaining results which previously Friedlander, Granville, Hildebrand and Maier [4] established conditionally on the Generalized Riemann Hypothesis. We will describe these in Section 5.…”

Section: Corollary 13 Let a S H F Q And γ Q Be As Above Let X Bsupporting

confidence: 57%

“…Our constraint on the small primes dividing is less restrictive than the corresponding condition there, though our localization of the x ± values is worse (in [4] the x ± values are localized in intervals (x/2, 2x)).…”

Section: Limitations To the Equidistribution Of Primesmentioning

confidence: 86%

“…Theorem 5.2 should be compared with Theorem A2 of [4]. Our bound is y −τ (1+o (1)) if y = exp((log ) τ ) with 0 < τ < 1/2.…”

Section: Theorem 52 Let Be Large and Suppose That (Log )mentioning

confidence: 99%

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An uncertainty principle for arithmetic sequences

Granville

Soundararajan

2007

Ann. Math.

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Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are "well-distributed" in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with the "uncertainty principle". In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples.

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Section: Corollary 13 Let a S H F Q And γ Q Be As Above Let X Bsupporting

confidence: 57%

Section: Limitations To the Equidistribution Of Primesmentioning

confidence: 86%

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An uncertainty principle for arithmetic sequences

Granville

Soundararajan

2007

Ann. Math.

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“…Two further propositions, Theorem 19 and Theorem 20, use the prime distribution results from the 'Distribution estimates on arithmetic functions' section to give asymptotics for certain sums involving sieve weights and the von Mangoldt function; they are established in the 'Multidimensional Selberg sieves' section. Theorems 22,24,26, and 28 use the asymptotics established in Theorems 19 and 20, in combination with Lemma 18, to give various criteria for bounding H m , which all involve finding sufficiently strong candidates for a variety of multidimensional variational problems; these theorems are proven in the 'Reduction to a variational problem' section. These variational problems are analysed in the asymptotic regime of large k in the 'Asymptotic analysis' section, and for small and medium k in the 'The case of small and medium dimension' section, with the results collected in Theorems 23,25,27,and 29.…”

Section: Organization Of the Papermentioning

confidence: 99%

Variants of the Selberg sieve, and bounded intervals containing many primes

Polymath¹

2014

Res Math Sci

130

172

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For any m ≥ 1, let H m denote the quantity lim inf n→∞ (p n+m − p n ). A celebrated recent result of Zhang showed the finiteness of H 1 , with the explicit bound H 1 ≤ 70, 000, 000. This was then improved by us (the Polymath8 project) to H 1 ≤ 4680, and then by Maynard to H 1 ≤ 600, who also established for the first time a finiteness result for H m for m ≥ 2, and specifically that H m m 3 e 4m . If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H 1 ≤ 12, improving upon the previous bound H 1 ≤ 16 of Goldston, Pintz, and Yıldırım, as well as the bound H m m 3 e 2m . In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H 1 ≤ 246 unconditionally and H 1 ≤ 6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h 1 , h 2 , h 3 ), there are infinitely many n for which at least two of n + h 1 , n + h 2 , n + h 3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the 'parity problem' argument of Selberg to show that the H 1 ≤ 6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound

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“…Also, let u = log x/ log y. Φ(x, y) is an important function of analytic number theory. Various estimates for Φ(x, y) have been given by several authors (see, [1], [4], [8], [9] [19], [24]), and has a variety of applications (see, [7], [8], [11], [14]). …”

Section: §1 Introductionmentioning

confidence: 99%