Matomäki, Kaisa and Radziwiłł, Maksym and Tao, Terence
(2017)
*Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges.*
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## Abstract

We show that the expected asymptotic for the sums ∑_(X<n≤2X)Λ(n)Λ(n+h), ∑_(X<n≤2X)d_k(n)d_l(n+h), and ∑_(X<n≤2X)Λ(n)d_k(n+h) hold for almost all h∈[−H,H], provided that X8/33+ε≤H≤X^(1−ε), with an error term saving on average an arbitrary power of the logarithm over the trivial bound. Previous work of Mikawa and Baier-Browning-Marasingha-Zhao covered the range H≥X^(1/3+ε). We also obtain an analogous result for ∑_nΛ(n)Λ(N−n). Our proof uses the circle method and some oscillatory integral estimates (following a paper of Zhan) to reduce matters to establishing some mean-value estimates for certain Dirichlet polynomials associated to "Type d_3" and "Type d_4" sums (as well as some other sums that are easier to treat). After applying Hölder's inequality to the Type d_3 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 d_4 sum is treated similarly using the classical L2 mean value theorem and the classical van der Corput exponential sum estimates.

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Additional Information: | KM was supported by Academy of Finland grant no. 285894. MR was supported by a NSERC Discovery Grant, the CRC program and a Sloan fellowship. TT 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. We are indebted to Yuta Suzuki for a reference and for pointing out a gap in the proof of Proposition 5.1 in an earlier version of the paper. We also thank Sary Drappeau for comments on the introduction. 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. | ||||||||||||||||||||

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Record Number: | CaltechAUTHORS:20180612-134458335 | ||||||||||||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20180612-134458335 | ||||||||||||||||||||

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ID Code: | 87015 | ||||||||||||||||||||

Collection: | CaltechAUTHORS | ||||||||||||||||||||

Deposited By: | Tony Diaz | ||||||||||||||||||||

Deposited On: | 12 Jun 2018 20:56 | ||||||||||||||||||||

Last Modified: | 03 Oct 2019 19:51 |

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