Abstract
We study asymptotics of sums of the form (Formula presented.), (Formula presented.), (Formula presented.), and ∑n Λ(n)Λ(N − n), where Λ is the von Mangoldt function, dk is the kth divisor function, and N, X are large. Our main result is that the expected asymptotic for the first three sums holds for almost all h ∈ [−H, H], provided that Xσ+ε ≤ H ≤ X1−ε for some ε > 0, where σ := 8/33 = 0.2424 ..., with an error term saving on average an arbitrary power of the logarithm over the trivial bound. This improves upon results of Mikawa and Baier–Browning–Marasingha–Zhao, who obtained statements of this form with σ replaced by 1/3. We obtain an analogous result for the fourth sum for most N in an interval of the form [X, X + H] with (Formula presented.). Our method starts with a variant of an argument from a paper of Zhan, using the circle method and some oscillatory integral estimates to reduce matters to establishing some mean-value estimates for certain Dirichlet polynomials associated to ‘Type d3’ and ‘Type d4’ sums (as well as some other sums that are easier to treat). After applying Hölder's inequality to the Type d3 sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. The Type d4 sum is treated similarly using the classical L2 mean value theorem and the classical van der Corput exponential sum estimates. In a sequel to this paper we will obtain related results for the correlations involving dk(n) for much smaller values of H but with weaker bounds.
Original language | English (US) |
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Pages (from-to) | 284-350 |
Number of pages | 67 |
Journal | Proceedings of the London Mathematical Society |
Volume | 118 |
Issue number | 2 |
DOIs | |
State | Published - Feb 1 2019 |
Funding
Received 5 July 2017; revised 26 June 2018; published online 31 July 2018. 2010 Mathematics Subject Classification 11N37 (primary). K. Matomäki was supported by Academy of Finland grant no. 285894. M. Radziwi l l was supported by a NSERC Discovery Grant, the CRC program and a Sloan fellowship. T. Tao was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment and by NSF grant DMS-1266164. Part of this paper was written while the authors were in residence at MSRI in Spring 2017, which is supported by NSF grant DMS-1440140.
ASJC Scopus subject areas
- General Mathematics