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Higher correlations of divisor sums related to primes, II: Variations of . . .
, 2007
"... We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the ..."
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We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet Lfunctions, we obtain an Ω±result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ωresults for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes.
Primes in Tuples I
"... We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the ElliottHalberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. E ..."
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We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the ElliottHalberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, pn+1 − pn lim inf =0. n→ ∞ log pn We will quantify this result further in a later paper (see (1.9) below).
Small gaps between prime numbers: the work of GoldstonPintzYıldırım
, 2000
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Polignac Numbers, Conjectures of Erdős on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture
, 2013
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Yıldırım, Small gaps between primes or almost primes
"... Abstract. Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that (pn+1 − pn) lim inf =0. n→ ∞ log pn We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of ex ..."
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Abstract. Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that (pn+1 − pn) lim inf =0. n→ ∞ log pn We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of exactly two distinct primes. We prove that lim inf n→ ∞ (qn+1 − qn) ≤ 26. If an appropriate generalization of the ElliottHalberstam Conjecture is true, then the above bound can be improved to 6. 1.
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.
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
Different Approaches to the Distribution of Primes
 MILAN JOURNAL OF MATHEMATICS
, 2009
"... In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction. ..."
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In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction.
Chen’s double sieve, Goldbach’s conjecture and the twin prime problem
 Institut Elie Cartan UMR 7502 UHPCNRSINRIA Université Henri Poincaré (Nancy 1) 54506 Vandœuvre–lès–Nancy FRANCE e–mail: wujie@iecn.unancy.fr
"... Abstract. For every even integer N, denote by D1,2(N) the number of representations of N as a sum of a prime and an integer having at most two prime factors. In this paper, we give a new lower bound for D1,2(N). ..."
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Abstract. For every even integer N, denote by D1,2(N) the number of representations of N as a sum of a prime and an integer having at most two prime factors. In this paper, we give a new lower bound for D1,2(N).