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Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges

Matomäki, Kaisa and Radziwiłł, Maksym and Tao, Terence (2019) Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges. Mathematische Annalen, 374 (1-2). pp. 793-840. ISSN 0025-5831. doi:10.1007/s00208-018-01801-4.

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We study the problem of obtaining asymptotic formulas for the sums ∑_(X<n≤2X)d_k(n)dl(n+h) and ∑_(X<n≤2X)Λ(n)d_k(n+h), where Λ is the von Mangoldt function, dk is the kth divisor function, X is large and k≥l≥2are real numbers. We show that for almost all h∈[−H,H] with H=(logX)^(10000k log k), the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of Λ(n)Λ(n+h) and we obtained better estimates for the error terms at the price of having to take H=X^(8/33+ε).

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Additional Information:© 2019 Springer-Verlag GmbH Germany, part of Springer Nature. Received: 31 January 2018; Revised: 27 December 2018; First Online: 21 January 2019. KM was supported by Academy of Finland Grant no. 285894. MR was supported by a NSERC DG 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. 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.
Funding AgencyGrant Number
Academy of Finland285894
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Canada Research Chairs ProgramUNSPECIFIED
Alfred P. Sloan FoundationUNSPECIFIED
Simons FoundationUNSPECIFIED
James and Carol Collins ChairUNSPECIFIED
Mathematical Analysis and Application Research Fund EndowmentUNSPECIFIED
Issue or Number:1-2
Record Number:CaltechAUTHORS:20180612-090354076
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Official Citation:Matomäki, K., Radziwiłł, M. & Tao, T. Math. Ann. (2019) 374: 793.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:86991
Deposited By: Tony Diaz
Deposited On:12 Jun 2018 16:15
Last Modified:15 Nov 2021 20:44

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