We give a correction to the previous paper of S. F. Cao and T. T. Wang (2021). Additionally, provide a conversion formula of the power mean of a certain exponential sums weighted by a Dirichlet character $ \chi_{3} $ and some of its applications.
Citation: Guohui Chen, Wenpeng Zhang. One kind two-term exponential sums weighted by third-order character[J]. AIMS Mathematics, 2024, 9(4): 9597-9607. doi: 10.3934/math.2024469
We give a correction to the previous paper of S. F. Cao and T. T. Wang (2021). Additionally, provide a conversion formula of the power mean of a certain exponential sums weighted by a Dirichlet character $ \chi_{3} $ and some of its applications.
[1] | T. M. Apostol, Introduction to Analytic Number Theory, New York: Springer-Verlag, 1976. |
[2] | S. F. Cao, T. T. Wang, On the hybrid power mean of two-term exponential sums and cubic Gauss sums, J. Math., 8 (2021), 6638156. https://doi.org/10.1155/2021/6638156 doi: 10.1155/2021/6638156 |
[3] | R. Duan, W. P. Zhang, On the fourth power mean of the generalized two-term exponential sums, Math. Rep., 22 (2020), 205–212. |
[4] | J. Greene, D. Stanton, The triplication formula for Gauss sums, Aequat. Math., 30 (1986), 143–141. https://doi.org/10.1007/BF02189920 doi: 10.1007/BF02189920 |
[5] | X. X. Li, Z. F. 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. https://doi.org/10.1186/1029-242X-2013-504 doi: 10.1186/1029-242X-2013-504 |
[6] | T. T. Wang, W. P. Zhang, On the eighth power mean of the two-term exponential sums, Finite Fields Appl., 92 (2023), 102285. https://doi.org/10.1016/j.ffa.2023.102285 doi: 10.1016/j.ffa.2023.102285 |
[7] | X. X. Wang, L. Wang, A new two-term exponential sums and its fourth power mean, Rendiconti Circolo Mate. Palermo, 8 (2023), 4041–4052. https://doi.org/10.1007/s12215-023-00879-3 doi: 10.1007/s12215-023-00879-3 |
[8] | J. M. Yu, R. J. Yuan, T. T. Wang, The fourth power mean value of one kind two-term exponential sums, AIMS Math., 9 (2022), 17045–17060. https://doi.org/10.3934/math.2022937 doi: 10.3934/math.2022937 |
[9] | H. Zhang, W. P. Zhang, The fourth power mean of two-term exponential sums and its application, Math. Rep. (Bucureşti), 96 (2017), 75–83. |
[10] | J. Zhang, W. P. Zhang, A certain two-term exponential sum and its fourth power means, AIMS Math., 6 (2020), 7500–7509. https://doi.org/10.3934/math.2020480 doi: 10.3934/math.2020480 |
[11] | W. P. Zhang, Moments of generalized quadratic Gauss sums weighted by $L$-functions, J. Number Theory, 2 (2002), 304–314. https://doi.org/10.1006/jnth.2001.2715 doi: 10.1006/jnth.2001.2715 |
[12] | W. P. Zhang, D. Han, On the sixth power mean of the two-term exponential sums, J. Number Theory, 136 (2014), 403–413. https://doi.org/10.1016/j.jnt.2013.10.022 doi: 10.1016/j.jnt.2013.10.022 |
[13] | W. P. Zhang, J. Y. Hu, The number of solutions of the diagonal cubic congruence equation $\bmod p$, Math. Rep., 20 (2018), 73–80. |
[14] | W. P. Zhang, H. L. Li, Elementary Number Theory, Xi'an: Shaanxi Normal University Press, 2013. |
[15] | W. P. Zhang, Y. Y. Meng, On the fourth power mean of one special two-term exponential sums, AIMS Math., 4 (2023), 8650–8660. https://doi.org/10.3934/math.2023434 doi: 10.3934/math.2023434 |
[16] | W. P. Zhang, Y. Y. Meng, On the sixth power mean of one kind two-term exponential sums weighted by Legendre's symbol modulo $p$, AIMS Math., 7 (2021), 6961–6974. https://doi.org/10.3934/math.2021408 doi: 10.3934/math.2021408 |