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AIMS Mathematics

2020, Volume 5, Issue 6: 7500-7509. doi: 10.3934/math.2020480
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Research article

A certain two-term exponential sum and its fourth power means

  • 1.
    School of Information Engineering, Xi'an University, Xi'an, Shaanxi, 710065, P. R. China
  • 2.
    School of Mathematics, Northwest University, Xi'an, Shaanxi, 710127, P. R. China
  • Received: 31 July 2020 Accepted: 18 September 2020 Published: 23 September 2020

  • MSC : 11L03, 11L05

  • The main purpose of this article is using the properties of the Legendre's symbol and the classical Gauss sums to study the calculating problem of the fourth power mean of a certain two-term exponential sums, and give an interesting calculating formula for it.

    Citation: Jin Zhang, Wenpeng Zhang. A certain two-term exponential sum and its fourth power means[J]. AIMS Mathematics, 2020, 5(6): 7500-7509. doi: 10.3934/math.2020480

    Related Papers:

  • The main purpose of this article is using the properties of the Legendre's symbol and the classical Gauss sums to study the calculating problem of the fourth power mean of a certain two-term exponential sums, and give an interesting calculating formula for it.


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    [1] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
    [2] W. P. Zhang, H. L. Li, Elementary Number Theory, Shaanxi Normal University Press, Xi'an, 2013.
    [3] W. Narkiewicz, Classical Problems in Number Theory, Polish Scientifc Publishers, WARSZAWA, 1986.
    [4] K. Ireland, M. Rosen, A classical introduction to modern number theory, Springer-Verlag, New York, 1982.
    [5] H. Zhang, W. P. Zhang, The fourth power mean of two-term exponential sums and its application, Mathematical Reports, 19 (2017), 75-81.
    [6] W. P. Zhang, D. Han, On the sixth power mean of the two-term exponential sums, J. Number Theory, 136 (2014), 403-413.
    [7] D. Han, A Hybrid mean value involving two-term exponential sums and polynomial character sums, Czechoslovak Mathematical J., 64 (2014), 53-62.
    [8] X. Y. Liu, W. P. Zhang, On the high-power mean of the generalized Gauss sums and Kloosterman sums, Mathematics, 7 (2019), 907.
    [9] X. X. Li, J. Y. Hu, The hybrid power mean quartic Gauss sums and Kloosterman sums, Open Math., 15 (2017), 151-156.
    [10] L. Chen, Z. Y. Chen, Some new hybrid power mean formulae of trigonometric sums, Advances Differences Equation, (2020) 2020: 220.
    [11] L. Chen, J. Y. Hu, A linear Recurrence Formula Involving Cubic Gauss Sums and Kloosterman Sums, Acta Mathematica Sinica (Chinese Series), 61 (2018), 67-72.
    [12] S. Chowla, J. Cowles, M. Cowles, On the number of zeros of diagonal cubic forms, J. Number Theory, 9 (1977), 502-506.
    [13] W. P. Zhang, J. Y. Hu, The number of solutions of the diagonal cubic congruence equation mod p, Mathematical Reports, 20 (2018), 70-76.
    [14] A. Weil, Basic number theory, Springer-Verlag, New York, 1974
    [15] A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A., 34 (1948), 203-210.
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  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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