This article applied the properties of character sums, quadratic character, and classical Gauss sums to study the calculations of the hybrid power mean of the generalized Gauss sums and the generalized two-term exponential sums. It also provided exact formulas for calculating these hybrid power means.
Citation: Xue Han, Tingting Wang. The hybrid power mean of the generalized Gauss sums and the generalized two-term exponential sums[J]. AIMS Mathematics, 2024, 9(2): 3722-3739. doi: 10.3934/math.2024183
This article applied the properties of character sums, quadratic character, and classical Gauss sums to study the calculations of the hybrid power mean of the generalized Gauss sums and the generalized two-term exponential sums. It also provided exact formulas for calculating these hybrid power means.
| [1] |
A. Weil, On some exponential sums, Proc. Nat. Acad. Sci., 34 (1948), 204–207. http://dx.doi.org/10.1073/pnas.34.5.204 doi: 10.1073/pnas.34.5.204
|
| [2] |
X. Lv, W. Zhang, A new hybrid power mean involving the generalized quadratic gauss sums and sums analogous to kloosterman sums, Lith. Math. J., 57 (2017), 359–366. http://dx.doi.org/10.1007/s10986-017-9366-z doi: 10.1007/s10986-017-9366-z
|
| [3] |
G. Djankovic, D. Dokic, N. Lela, I. Vrecica, On some hybrid power moments of products of generalized quadratic Gauss sums and Kloosterman sums, Lith. Math. J., 58 (2018), 1–14. http://dx.doi.org/10.1007/s10986-018-9383-6 doi: 10.1007/s10986-018-9383-6
|
| [4] | X. Li, C. Wu, Generalized quadratic Gauss sums and mixed means of generalized Kloosterman sums (Chinese), Mathematics in Practics and Theory, 49 (2019), 236–242. |
| [5] |
X. Liu, Y. Meng, On the fourth hybrid power mean involving the generalized gauss sums, J. Math., 2023 (2023), 8732814. http://dx.doi.org/10.1155/2023/8732814 doi: 10.1155/2023/8732814
|
| [6] |
X. Li, J. Hu, The hybrid power mean quartic Gauss sums and Kloosterman sums, Open Math., 15 (2017), 151–156. http://dx.doi.org/10.1515/math-2017-0014 doi: 10.1515/math-2017-0014
|
| [7] |
X. Li, Z. Xu, The fourth power mean of the generalized two-term exponential sums and its upper and lower bound estimates, J. Inequal. Appl., 2013 (2013), 504. http://dx.doi.org/10.1186/1029-242X-2013-504 doi: 10.1186/1029-242X-2013-504
|
| [8] |
X. Li, The hybrid power mean of the quartic Gauss sums and the two-term exponential sums, Adv. Differ. Equ., 2018 (2018), 236. http://dx.doi.org/10.1186/s13662-018-1658-z doi: 10.1186/s13662-018-1658-z
|
| [9] |
D. Han, A hybrid mean value involving two-term exponential sums and polynomial character sums, Czech. Math. J., 64 (2014), 53–62. http://dx.doi.org/10.1007/s10587-014-0082-0 doi: 10.1007/s10587-014-0082-0
|
| [10] |
L. Chen, X. Wang, A new fourth power mean of two-term exponential sums, Open Math., 17 (2019), 407–414. http://dx.doi.org/10.1515/math-2019-0034 doi: 10.1515/math-2019-0034
|
| [11] |
W. Zhang, D. Han, On the sixth power mean of the two-term exponential sums, J. Number Theory, 136 (2014), 403–413. http://dx.doi.org/10.1016/j.jnt.2013.10.022 doi: 10.1016/j.jnt.2013.10.022
|
| [12] | L. Hua, Introduction to number theory, Beijing: Science Press, 1979. |
| [13] | T. Apostol, Introduction to analytic number theory, New York: Springer-Verlag, 1976. http://dx.doi.org/10.1007/978-1-4757-5579-4 |
| [14] | W. Zhang, H. Li, Elementary number theory, Xi'an: Shaanxi Normal University Press, 2008. |
| [15] | K. Ireland, M. Rosen, A classical introduction to modern number theory, New York: Springer-Verlag, 1990. http://dx.doi.org/10.1007/978-1-4757-2103-4 |
| [16] |
X. Lv, W. Zhang, The generalized quadratic Gauss sums and its sixth power mean, AIMS Mathematics, 6 (2021), 11275–11285. http://dx.doi.org/10.3934/math.2021654 doi: 10.3934/math.2021654
|
Xue Han, Tingting Wang. The hybrid power mean of the generalized Gauss sums and the generalized two-term exponential sums[J]. AIMS Mathematics, 2024, 9(2): 3722-3739. doi: 10.3934/math.2024183
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