Results 1  10
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18
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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Cited by 28 (6 self)
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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.
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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Cited by 8 (2 self)
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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 between ..."
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Cited by 7 (3 self)
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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 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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Cited by 6 (1 self)
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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).
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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Cited by 3 (0 self)
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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).
SMALL GAPS BETWEEN PRIMES II (PRELIMINARY)
"... Abstract. We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (pn+1 − pn) (1.1) lim inf n→ ∞ log pn(log log pn) −1 < ∞. log log log log pn Further we show that supposing the ..."
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Cited by 2 (2 self)
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Abstract. We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (pn+1 − pn) (1.1) lim inf n→ ∞ log pn(log log pn) −1 < ∞. log log log log pn Further we show that supposing the validity of the Bombieri–Vinogradov theorem up to Q ≤ Xϑ with any level ϑ>1/2 we have bounded differences between consecutive primes infinitely often: (1.2) lim inf n→ ∞ (pn+1 − pn) ≤ C(ϑ) with a constant C(ϑ) depending only on ϑ. If the Bombieri–Vinogradov theorem holds with a level ϑ>20/21, in particular if the Elliott–Halberstam conjecture holds, then we obtain (1.3) lim inf n→ ∞ (pn+1 − pn) ≤ 20, that is pn+1 − pn ≤ 20 for infinitely many n. Inequalities (1.2)–(1.3) will follow from the even stronger following result Theorem A. Suppose the Bombieri–Vinogradov theorem is true for Q ≤ Xϑ with some ϑ>1/2. Then there exists a constant C ′ (ϑ) such that any admissible ktuple contains at least two primes for any (1.4) k ≥ C ′ (ϑ) if ϑ>1/2, where C ′ (ϑ) is an explicitly calculable constant depending only on ϑ. Further we have at least two primes for (1.5) k =7 if ϑ>20/21. Remark. For the definition of admissibility see (2.2) below. We will show some more general results for the quantity (ν is a given positive integer) (1.6) Eν = lim inf n→∞ pn+ν − pn log pn
Prime pairs and zeta’s zeros
, 2007
"... Abstract. There is extensive numerical support for the primepair conjecture (PPC) of Hardy and Littlewood (1923) on the asymptotic behavior of π2r(x), the number of prime pairs (p, p + 2r) with p ≤ x. However, it is still not known whether there are infinitely many prime pairs with given even diffe ..."
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Cited by 2 (2 self)
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Abstract. There is extensive numerical support for the primepair conjecture (PPC) of Hardy and Littlewood (1923) on the asymptotic behavior of π2r(x), the number of prime pairs (p, p + 2r) with p ≤ x. However, it is still not known whether there are infinitely many prime pairs with given even difference! Using a strong hypothesis on (weighted) equidistribution of primes in arithmetic progressions, Goldston, Pintz and Yildirim have recently shown that there are infinitely many pairs of primes differing by at most sixteen. The present author uses a Tauberian approach to derive that the PPC is equivalent to specific boundary behavior of certain functions involving zeta’s complex zeros. Under Riemann’s Hypothesis (RH) and on the real axis these functions resemble paircorrelation expressions. A speculative extension of Montgomery’s classical work (1973) would imply that there must be an abundance of prime pairs. 1.