Higher correlations of divisor sums related to primes II: variations of the error term in the prime number theorem

By D. Goldston, C. Y. Yıldırım. 2004

55 citations recorded by the source corpus.

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 DirichletL‐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 λR(n) and ΛR(n) can be employed in diverse problems concerning primes.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fdafd204e93cf7ac740b24c3d3af33d594c6bb5a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

The primes contain arbitrarily long arithmetic progressions (2004, result): “Since the correlation condition is an upper bound rather than an asymptotic, it is fairly easy to obtain; we shall prove it using the arguments of Goldston and Yıldırım (since we are using those methods in any case to prove the linear forms condition), but these upper bounds could also be obtained b”
Decompositions, approximate structure, transference, and the Hahn–Banach theorem (2008, methodology): “) The construction of ν was based on very recent (at the time) results of Goldston and Yıldırım [GY].”
Small gaps between primes (2013, methodology): “The key new idea in the paper of Goldston, Pintz and Yıldırım [5] was to consider more general sieve weights of the form for a suitable smooth function F .”
The primes contain arbitrarily long polynomial progressions (2006, methodology): “(33) This differs slightly from the majorant introduced by Goldston and Yıldırım in [14] and used in [18]; in our notation, the majorants from those papers corresponds to the case χ(t):=max(1−|t|, 0).”
The dichotomy between structure and randomness, arithmetic progressions, and the primes (2005, methodology): “One still needs an enveloping sieve ν, but instead of using a Selberg-type sieve that enjoys good Fourier coefficient control, it turns out to be more convenient to use an enveloping sieve 7 of Goldston and Yıldırım [15], [16], [17] which has good control on k-point correlations (indeed, it behaves ”