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

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Abstract

We study the problem of obtaining asymptotic formulas for the sums \(\sum _{X < n \le 2X} d_k(n) d_l(n+h)\) and \(\sum _{X < n \le 2X} \Lambda (n) d_k(n+h)\), where \(\Lambda \) is the von Mangoldt function, \(d_k\) is the \(k^{{\text {th}}}\) divisor function, X is large and \(k \ge l \ge 2\) are integers. We show that for almost all \(h \in [-H, H]\) with \(H = (\log X)^{10000 k \log k}\), the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of \(\Lambda (n) \Lambda (n + h)\) and we obtained better estimates for the error terms at the price of having to take \(H = X^{8/33 + \varepsilon }\).

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Notes

  1. The results here can also be applied to the sums \(\sum _{n \le X} d_k(n) d_l(n+h)\) and \(\sum _{n \le X} \Lambda (n) d_k(n+h)\) after some minor modifications to the coefficients of the polynomials \(P_{k,l,h}\) and \(Q_{k,h}\); we leave the details to the interested reader.

  2. One may also invoke truncated divisor sum moment estimates [3, 21] here.

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Turku, 20014, Turku, Finland

    Kaisa Matomäki

  2. Department of Mathematics, Caltech, 1200 E California Blvd, Pasadena, CA, 91125, USA

    Maksym Radziwiłł

  3. Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA, 90095, USA

    Terence Tao

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  1. Kaisa Matomäki
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  2. Maksym Radziwiłł
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  3. Terence Tao
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Correspondence to Maksym Radziwiłł.

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Communicated by Kannan Soundararajan.

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Matomäki, K., Radziwiłł, M. & Tao, T. Correlations of the von Mangoldt and higher divisor functions II: divisor correlations in short ranges. Math. Ann. 374, 793–840 (2019). https://doi.org/10.1007/s00208-018-01801-4

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