On the sixth power mean values of a generalized two-term exponential sums

Shujie Zhou; Li Chen; Shujie Zhou; Li Chen

doi:10.3934/math.20231438

AIMS Mathematics

2023, Volume 8, Issue 11: 28105-28119. doi: 10.3934/math.20231438

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On the sixth power mean values of a generalized two-term exponential sums

Shujie Zhou ¹,
Li Chen ^{2
,
,}

1.
School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, China
2.
School of Mathematics, Xi'an University of Finance and Economics, Xi'an, China

Received: 15 August 2023 Revised: 02 September 2023 Accepted: 12 September 2023 Published: 12 October 2023
MSC : 11L03, 11L05

This paper examines the evaluations of sixth power mean values of a generalized two-term exponential sums. In the case $ p\equiv 3\bmod 4 $, we try to establish two precise formulas by applying the properties of character sums and the number of the solutions of relevant congruence equations modulo an odd prime $ p $.
- the two-term exponential sums,
- sixth power mean values,
- elementary method,
- congruence equation,
- calculating formula
Citation: Shujie Zhou, Li Chen. On the sixth power mean values of a generalized two-term exponential sums[J]. AIMS Mathematics, 2023, 8(11): 28105-28119. doi: 10.3934/math.20231438

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Abstract

This paper examines the evaluations of sixth power mean values of a generalized two-term exponential sums. In the case $ p\equiv 3\bmod 4 $, we try to establish two precise formulas by applying the properties of character sums and the number of the solutions of relevant congruence equations modulo an odd prime $ p $.

References

[1]	H. Zhang, W. P. Zhang, The fourth power mean of two-term exponential sums and its application, Math. Rep., 19 (2017), 75–81.
[2]	R. Duan, W. P. Zhang, On the fourth power mean of the generalized two-term exponential sums, Math. Rep., 22 (2020), 205–212.
[3]	W. P. Zhang, Y. Y. Meng, On the sixth power mean of the two-term exponential sums, Acta Math. Sin. English Ser., 38 (2022), 510–518. https://doi.org/10.1007/s10114-022-0541-8 doi: 10.1007/s10114-022-0541-8
[4]	L. Chen, X. Wang, A new fourth power mean of two-term exponential sums, Open Math., 17 (2019), 407–414. https://doi.org/10.1515/math-2019-0034 doi: 10.1515/math-2019-0034
[5]	X. Y. Liu, W. P. Zhang, On the high-power mean of the generalized Gauss sums and Kloosterman sums, Mathematics, 7 (2019), 907. https://doi.org/10.3390/math7100907 doi: 10.3390/math7100907
[6]	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
[7]	H. N. Liu, W. M. Li, On the fourth power mean of the three-term exponential sums, Adv. Math., 46 (2017), 529–547.
[8]	Y. H. Yu, W. P. Zhang, On the sixth power mean value of the generalized three-term exponential sums, Abstract Appl. Anal., 2014 (2014), 474726. https://doi.org/10.1155/2014/474726 doi: 10.1155/2014/474726
[9]	X. C. Du, X. X. Li, On the fourth power mean of generalized three-term exponential sums, J. Math. Res. Appl., 35 (2015), 92–96.
[10]	X. Y. Wang, X. X. Li, One kind sixth power mean of the three-term exponential sums, Open Math., 15 (2017), 705–710. https://doi.org/10.1515/math-2017-0060 doi: 10.1515/math-2017-0060
[11]	H. S. Kim, T. Kim, On certain values of $p$-adic $q$-$L$-functions, Rep. Fac. Sci. Engrg. Saga Univ. Math., 23 (1995), 1–7.
[12]	T. Kim, On explicit formulas of $p$-adic $q$-$L$-functions, Kyushu J. Math., 48 (1994), 73–86.
[13]	T. M. Apostol, Introduction to Analytic Number Theory, New York: Springer, 1976.
[14]	K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, New York: Springer, 1982.
[15]	W. P. Zhang, H. L. Li, Elementary Number Theory, Xi'an: Shaanxi Normal University Press, 2013.

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