Abstract
The main purpose of this article is to study the calculating problem of the sixth power mean of the two-term exponential sums, and give an interesting calculating formula for it. At the same time, the paper also provides a new and effective method for the study of the high order power mean of the exponential sums.
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References
Apostol, T. M.: Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976
Berndt, B. C., Evans, R. J.: The determination of Gauss sums. Bulletin of the American Mathematical Society, 5, 107–128 (1981)
Chen, L., Chen, Z. Y.: Some new hybrid power mean formulae of trigonometric sums. Advances in Differences Equation, 2020, Paper No. 220, 9 pp. (2020)
Chen, L., Hu, J. Y.: A linear recurrence formula involving cubic Gauss sums and Kloosterman sums. Acta Mathematica Sinica, Chinese Series, 61, 67–72 (2018)
Chowla, S., Cowles, J., Cowles, M.: On the number of zeros of diagonal cubic forms. Journal of Number Theory, 9, 502–506 (1977)
Han, D.: A Hybrid mean value involving two-term exponential sums and polynomial character sums. Czechoslovak Mathematical Journal, 64, 53–62 (2014)
Ireland, K., Rosen, M.: A classical introduction to modern number theory, Springer-Verlag, New York, 1982
Li, X. X., Hu, J. Y.: The hybrid power mean quartic Gauss sums and Kloosterman sums. Open Mathematics, 15, 151–156 (2017)
Liu, X. Y., Zhang, W. P.: On the high-power mean of the generalized Gauss sums and Kloosterman sums. Mathematics, 7, Paper No. 907, 9 pp. (2019)
Zhang, H., Zhang, W. P.: The fourth power mean of two-term exponential sums and its application. Mathematical Reports, 19, 75–81 (2017)
Zhang, J., Zhang, W. P.: A certain two-term exponential sum and its fourth power means. AIMS Mathematics, 5(6), 7500–7509 (2020)
Zhang, W. P., Han, D.: On the sixth power mean of the two-term exponential sums. Journal of Number Theory, 136, 403–413 (2014)
Zhang, W. P., Hu, J. Y.: The number of solutions of the diagonal cubic congruence equation mod p. Mathematical Reports, 20, 70–76 (2018)
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The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.
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Supported by NSFC (Grant No. 11771351) and NSBRP (Grant No. 2019JM-207)
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Zhang, W.P., Meng, Y.Y. On the Sixth Power Mean of the Two-term Exponential Sums. Acta. Math. Sin.-English Ser. 38, 510–518 (2022). https://doi.org/10.1007/s10114-022-0541-8
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DOI: https://doi.org/10.1007/s10114-022-0541-8