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Mathematische Annalen

, Volume 374, Issue 1–2, pp 793–840 | Cite as

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

  • Kaisa Matomäki
  • Maksym RadziwiłłEmail author
  • Terence Tao
Article
  • 85 Downloads

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 \(H = X^{8/33 + \varepsilon }\).

Notes

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.

References

  1. 1.
    Baier, S., Browning, T.D., Marasingha, G., Zhao, L.: Averages of shifted convolutions of \(d_3(n)\). Proc. Edinb. Math. Soc. (2) 55(3), 551–576 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brüdern, J.: Binary additive problems and the circle method, multiplicative sequences and convergent sieves, analytic number theory, 91–132. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar
  3. 3.
    Bourgain, J.: On \(\Lambda (p)\)-subsets of squares. Isr. J. Math. 67, 291–311 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Conrey, J.B., Gonek, S.M.: High moments of the Riemann zeta-function. Duke Math. J. 107, 577–604 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    de la Bretèche, R., Granville, A.: Densité des friables. Bulletin de la Société Mathématique de France 142(2), 303–348 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    de la Bretèche, R., Fiorilli, D.: Major arcs and moments of arithmetical sequences (pre-print). arxiv:1611.08312
  7. 7.
    Ford, K., Halberstam, H.: The Brun–Hooley sieve. J. Number Theory 81(2), 335–350 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gallagher, P.X.: A large sieve density estimate near \(\sigma = 1\). Invent. Math. 11, 329–339 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Goldston, D.A., Yıldırım, C.Y.: Higher correlations of divisor sums related to primes, III: \(k\)-correlations (preprint, available at AIM preprints) Google Scholar
  10. 10.
    Green, B., Tao, T.: Restriction theory of the Selberg sieve, with applications. J. Théor. Nombres Bordeaux 18(1), 147–182 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Green, B., Tao, T.: Linear equations in primes. Ann. Math. (2) 171(3), 1753–1850 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Henriot, K.: Nair–Tenenbaum bounds uniform with respect to the discriminant. Math. Proc. Camb. Philos. Soc. 152(3), 405–424 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Henriot, K.: Nair–Tenenbaum uniform with respect to the discriminant—erratum. Math. Proc. Camb. Philos. Soc. 157(2), 375–377 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ivić, A.: The General Additive Divisor Problem and Moments of the Zeta-Function. New Trends in Probability and Statistics (Palanga, 1996), vol. 4, pp. 69–89. VSP, Utrecht (1997)zbMATHGoogle Scholar
  15. 15.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory, vol. 53. Colloquium Publications, American Mathematical Society, Providence (2004)zbMATHGoogle Scholar
  16. 16.
    Matomäki, K., Radziwiłł, M.: Multiplicative functions in short intervals. Ann. Math. (2) 183(3), 1015–1056 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Matomäki, K., Radziwiłł, M., Tao, T.: Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges. Proc. Lond. Math. Soc. (2018).  https://doi.org/10.1112/plms.12181
  18. 18.
    Matthiesen, L.: Correlations of the divisor function. Proc. Lond. Math. Soc. (3) 104(4), 827–858 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Matthiesen, L.: Linear correlations of multiplicative functions (preprint) Google Scholar
  20. 20.
    Mikawa, H.: On prime twins. Tsukuba J. Math. 15, 19–29 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ruzsa, I.Z.: On an additive property of squares and primes. Acta Arith. 49, 281–289 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shiu, P.: A Brun–Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313, 161–170 (1980)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Tenenbaum, G.: Introduction to Analytic and Probabilistic Number Theory, Translated from the second French edition (1995) by C. B. Thomas. Cambridge Studies in Advanced Mathematics, vol. 46. Cambridge University Press, Cambridge (1995)Google Scholar
  24. 24.
    Vinogradov, A.I.: The \(SL_n\)-technique and the density hypothesis (in Russian). Zap. Naučn. Sem. LOMI AN SSSR 168, 5–10 (1988)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Kaisa Matomäki
    • 1
  • Maksym Radziwiłł
    • 2
    Email author
  • Terence Tao
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  2. 2.Department of MathematicsCaltechPasadenaUSA
  3. 3.Department of MathematicsUCLALos AngelesUSA

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