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Small gaps between primes
"... ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing betwe ..."
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ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between primes. 4 1.
Variants of the Selberg sieve, and bounded intervals containing many primes
, 2014
"... michaelnielsen.org/polymath1/ index.php?title=Bounded_gaps_ between_primes, Full list of author information is available at the end of the article For any m ě 1, let Hm denote the quantity lim infnÑ8ppn`m ´ pnq, where pn is the nth prime. A celebrated recent result of Zhang showed the finiteness of ..."
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michaelnielsen.org/polymath1/ index.php?title=Bounded_gaps_ between_primes, Full list of author information is available at the end of the article For any m ě 1, let Hm denote the quantity lim infnÑ8ppn`m ´ pnq, where pn is the nth prime. A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1 ď 70000000. This was then improved by us (the Polymath8 project) to H1 ď 4680, and then by Maynard to H1 ď 600, who also established for the first time a finiteness result for Hm for m ě 2, and specifically that Hm! m3e4m. If one also assumes the ElliottHalberstam conjecture, Maynard obtained the bound H1 ď 12, improving upon the previous bound H1 ď 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 ď 246 unconditionally, and H1 ď 6 under the assumption of the generalized ElliottHalberstam conjecture. Indeed, under the latter conjecture we show the stronger statement that for any admissible triple ph1, h2, h3q, 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 ď 6 bound is the best possible that one can obtain from purely sievetheoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound Hm! mep4 ´ 28157 qm, or Hm! me2m under the assumption of the ElliottHalberstam conjecture. We also obtain explicit upper bounds for Hm when m “ 2, 3, 4, 5.
NORM FORMS FOR ARBITRARY NUMBER FIELDS AS PRODUCTS OF LINEAR POLYNOMIALS
"... Abstract. Given a number field K/Q and a polynomial P ∈ Q[t], all of whose roots are in Q, let X be the variety defined by the equation NK(x) = P (t). Combining additive combinatorics with descent we show that the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak appr ..."
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Abstract. Given a number field K/Q and a polynomial P ∈ Q[t], all of whose roots are in Q, let X be the variety defined by the equation NK(x) = P (t). Combining additive combinatorics with descent we show that the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of X. Contents
ON THE OPTIMAL WEIGHT FUNCTION IN THE GOLDSTONPINTZYILDIRIM METHOD FOR FINDING SMALL GAPS BETWEEN CONSECUTIVE PRIMES
"... Abstract. We work out the optimization problem, initiated by K. Soundararajan, for the choice of the underlying polynomial P used in the construction of the weight function in the Goldston–Pintz–Yıldırım method for finding small gaps between primes. First we reformulate to a maximization problem on ..."
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Abstract. We work out the optimization problem, initiated by K. Soundararajan, for the choice of the underlying polynomial P used in the construction of the weight function in the Goldston–Pintz–Yıldırım method for finding small gaps between primes. First we reformulate to a maximization problem on L2 [0, 1] for a selfadjoint operator T, the norm of which is then the maximal eigenvalue of T. To find eigenfunctions and eigenvalues, we derive a differential equation which can be explicitly solved. The aimed maximal value is S(k) = 4/(k + ck1/3), achieved by the k − 1st integral of x1−k/2 √ Jk−2(α1 x), where α1 ∼ ck1/3 is the first positive root of the k − 2nd Bessel function Jk−2. As this naturally gives rise to a number of technical problems in the application of the GPY method, we also construct a polynomial P which is a simpler function yet it furnishes an approximately optimal extremal quantity, 4/(k + Ck1/3) with some other constant C. In the forthcoming paper of J. Pintz [8] it is indeed shown how this quasioptimal choice of the polynomial in the weight finally can exploit the GPY method to its theoretical limits. 1.
BOUNDED GAPS BETWEEN PRIMES IN NUMBER FIELDS AND FUNCTION FIELDS
"... Abstract. The Hardy–Littlewood prime ktuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we ext ..."
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Abstract. The Hardy–Littlewood prime ktuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the MaynardTao method to both number fields and the function field Fq(t).
PRIMES IN INTERVALS OF BOUNDED LENGTH
"... Abstract. In April 2013, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs of distinct primes which differ by no more than B. This is a massive breakthrough, makes the twin prime conjecture look highly plausible (which can be reinterpreted as the conjec ..."
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Abstract. In April 2013, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs of distinct primes which differ by no more than B. This is a massive breakthrough, makes the twin prime conjecture look highly plausible (which can be reinterpreted as the conjecture that one can take B 2) and his work helps us to better understand other delicate questions about prime numbers that had previously seemed intractable. The original purpose of this talk was to discuss Zhang’s extraordinary work, putting it in its context in analytic number theory, and to sketch a proof of his theorem. Zhang had even proved the result with B 70 000 000. Moreover, a cooperative team, polymath8, collaborating only online, had been able to lower the value of B to 4680. Not only had they been more careful in several difficult arguments in Zhang’s original paper, they had also developed Zhang’s techniques to be both more powerful and to allow a much simpler proof. Indeed the proof of Zhang’s Theorem, that will be given in the writeup of this talk, is based on these developments. In November, inspired by Zhang’s extraordinary breakthrough, James Maynard dra
Bounded length intervals containing two primes and an almostprime. Preprint, available at http://arxiv.org/abs/1205.5020
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THE DISTRIBUTION OF PRIME NUMBERS
, 2006
"... What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument. Lecture 1: The Cramér model and gaps between consec ..."
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What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument. Lecture 1: The Cramér model and gaps between consecutive primes The prime number theorem tells us that π(x), the number of primes below x, is ∼ x / logx. Equivalently, if pn denotes the nth smallest prime number then pn ∼ n log n. What is the distribution of the gaps between consecutive primes, pn+1 − pn? We have just seen that pn+1 − pn is approximately log n “on average”. How often do we get a gap of size 2 logn, say; or of size 1 log n? One way to make this question precise 2 is to fix an interval [α, β] (with 0 ≤ α < β) and ask for