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Harald Cramér and the distribution of prime numbers
 Scandanavian Actuarial J
, 1995
"... “It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and MittagLeffler. His works then, and later in the ..."
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Cited by 42 (2 self)
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“It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and MittagLeffler. His works then, and later in the midthirties, have had a profound influence on the way mathematicians think about the distribution of prime numbers. In this article, we shall focus on how Cramér’s ideas have directed and motivated research ever since. One can only fully appreciate the significance of Cramér’s contributions by viewing his work in the appropriate historical context. We shall begin our discussion with the ideas of the ancient Greeks, Euclid and Eratosthenes. Then we leap in time to the nineteenth century, to the computations and heuristics of Legendre and Gauss, the extraordinarily analytic insights of Dirichlet and Riemann, and the crowning glory of these ideas, the proof the “Prime Number Theorem ” by Hadamard and de la Vallée Poussin in 1896. We pick up again in the 1920’s with the questions asked by Hardy and Littlewood,
Variants of the Selberg sieve, and bounded intervals containing many primes
, 2014
"... For any m ≥ 1, let Hm denote the quantity lim infn→∞(pn+m − pn). A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1 ≤ 70, 000, 000. This was then improved by us (the Polymath8 project) to H1 ≤ 4680, and then by Maynard to H1 ≤ 600, who also established for ..."
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For any m ≥ 1, let Hm denote the quantity lim infn→∞(pn+m − pn). A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1 ≤ 70, 000, 000. This was then improved by us (the Polymath8 project) to H1 ≤ 4680, and then by Maynard to H1 ≤ 600, who also established for the first time a finiteness result for Hm form ≥ 2, and specifically that Hm m3e4m. If one also assumes the ElliottHalberstam conjecture, Maynard obtained the bound H1 ≤ 12, improving upon the previous bound H1 ≤ 16 of Goldston, Pintz, and Yıldırım, as well as the bound Hm m3e2m. 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 H1 ≤ 246 unconditionally and H1 ≤ 6 under the assumption of the generalized ElliottHalberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h1, h2, h3), there are infinitely many n for which at least two of n + h1, n + h2, n + h3 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 H1 ≤ 6 bound is the best possible that one can obtain from purely sievetheoretic considerations. For largerm, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound Hm me
Surpassing the Ratios Conjecture in the 1level density of Dirichlet Lfunctions
 ALGEBRA & NUMBER THEORY
, 2012
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