On a symmetric Diophantine equation with reciprocals

  • S. V. Konyagin
  • M. A. Korolev


An asymptotic formula is obtained for the number of solutions to a symmetric Diophantine equation with reciprocals, and its applications to problems related to the distribution of values of short Kloosterman sums are presented.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. A. Karatsuba, “Analogues of Kloosterman sums, ” Izv. Ross. Akad. Nauk, Ser. Mat. 59 (5), 93–102 (1995) [Izv. Math. 59, 971–981 (1995)].MathSciNetGoogle Scholar
  2. 2.
    P. V. Snurnitsyn, “On estimating the mean value of a short Kloosterman sum, ” Uch. Zap. Orlov. Gos. Univ., Ser.: Estestv. Tekh. Med. Nauki, No. 6-2, 212–215 (2012).Google Scholar
  3. 3.
    J. Bourgain and M. Z. Garaev, “Sumsets of reciprocals in prime fields and multilinear Kloosterman sums, ” Izv. Ross. Akad. Nauk, Ser. Mat. 78 (4), 19–72 (2014) [Izv. Math. 78, 656–707 (2014)].MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D. R. Heath-Brown, “The density of rational points on Cayley’s cubic surface, ” in Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonn, 2002 (Univ. Bonn, Bonn, 2003), Bonner Math. Schriften 360, 33 p.Google Scholar
  5. 5.
    K. K. Mardzhanishvili, “Estimate for an arithmetic sum, ” Dokl. Akad. Nauk SSSR 22 (7), 391–393 (1939).Google Scholar
  6. 6.
    I. S. Timergaliev, “Value distributions of analogues of Kloosterman’s sums, ” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 5, 37–41 (2013) [Moscow Univ. Math. Bull. 68 (5), 249–252 (2013)].MathSciNetzbMATHGoogle Scholar
  7. 7.
    R. N. Boyarinov, “Probabilistic methods in number theory and applications to the theory of the argument of the Riemann zeta function, ” Doctoral (Phys.–Math.) Dissertation (Moscow State Univ., Moscow, 2012).Google Scholar
  8. 8.
    R. N. Boyarinov, “On the rate of convergence of distributions of random variables, ” Dokl. Akad. Nauk 435 (3), 295–297 (2010) [Dokl. Math. 82 (3), 896–898 (2010)].MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations