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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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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.
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).
On Partitions of Goldbach’s Conjecture
, 2000
"... An approximate formula for the partitions of Goldbach’s Conjecture is derived using Prime Number Theorem and a probabilistic approach. A strong form of Goldbach’s conjecture follows in the form of a lower bounding function for the partitions of Goldbach’s conjecture. Numerical computations suggest t ..."
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An approximate formula for the partitions of Goldbach’s Conjecture is derived using Prime Number Theorem and a probabilistic approach. A strong form of Goldbach’s conjecture follows in the form of a lower bounding function for the partitions of Goldbach’s conjecture. Numerical computations suggest that the lower and upper bounding functions for the partitions satisfy a simple functional equation. Assuming that this invariant scaling property holds for all even integer n, the lower and upper bounds can be expressed as simple exponentials. 1 Goldbach’s Conjecture and Recent Progress Goldbach’s Conjecture states that every even integer> 2 can be expressed as a sum of two primes. The proof remains an unsolved problem since Goldbach first wrote the conjecture in a letter to Euler in 1792. However, significant progress has been made in recent years. On the front of verifying Goldbach’s Conjecture, no counterexample has