Higher correlations of divisor sums related to primes III: small gaps between primes

Abstract

We calculate the triple correlations for the truncated divisor sum λ R (n). The λ R (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 Λ R (n). However, when λ R (n) is used, the error in the singular series approximation is often much smaller than what Λ R (n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions, 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 Ω-results 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 λ R (n) and Λ R (n) can be employed in diverse problems concerning primes.

Small gaps between primes or almost primes

“….. Maier's method had the shortcoming that it produced a sparse set of gaps; prior authors had shown that small gaps occur in a positive proportion of all cases. Goldston and Yıldırım [9] proved the upper bound of 0.25 for a positive proportion of gaps. Recently, the first, third and fourth authors proved a best possible result in this direction.…”

From harmonic analysis to arithmetic combinatorics

“…Green and Tao in [85] credit Andrew Granville for pointing them to the preprint [72]. The correlation estimates therein, with relatively minor changes, were sufficient to prove that ν was pseudorandom in the sense of [85].…”

Restriction theory of the Selberg sieve, with applications

“…Objects of this type were extensively analysed by Goldston and Yıldırım [11,12,13], and in particular they saw how to asymptotically evaluate certain correlations of the form (8.1) provided R N cm . This was a crucial ingredient in our work in [18], but we should remark that other aspects of the function β R were investigated much earlier, and indeed β R was known to Selberg.…”

“…We note that results along the lines of Theorem 1.2 can be approached using sieve methods and the Hardy-Littlewood circle method in a more classical guise. For example, Tolev [33] showed that there are infinitely many 3-term progressions p 1 < p 2 < p 3 of primes such that p i + 2 is a product of at most r i primes, where (r 1 , r 2 , r 3 ) can be taken to be (5,5,8) or (4,5,11).…”