On two-term exponential sums and their mean values

Jianghua Li; Zepeng Zhang; Tianyu Pan; Jianghua Li; Zepeng Zhang; Tianyu Pan

doi:10.3934/era.2023282

Electronic Research Archive

2023, Volume 31, Issue 9: 5559-5572. doi: 10.3934/era.2023282

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On two-term exponential sums and their mean values

College of Sciences, Xi'an University of Technology, Xi'an, Shaanxi, China

Received: 23 January 2023 Revised: 21 April 2023 Accepted: 25 July 2023 Published: 14 August 2023

The mean value problems of exponential sums play a very important role in the research of analytic number theory, and many famous number theory problems are closely related to them. The main purpose of this paper is using some elementary methods and the number of the solutions of the congruence equations to study the calculating problem of some fourth power means of two-term exponential sums, and give exact calculating formulae and asymptotic formula for them.
- the two-term exponential sums,
- fourth power mean,
- elementary method,
- calculating formula
Citation: Jianghua Li, Zepeng Zhang, Tianyu Pan. On two-term exponential sums and their mean values[J]. Electronic Research Archive, 2023, 31(9): 5559-5572. doi: 10.3934/era.2023282

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Abstract

The mean value problems of exponential sums play a very important role in the research of analytic number theory, and many famous number theory problems are closely related to them. The main purpose of this paper is using some elementary methods and the number of the solutions of the congruence equations to study the calculating problem of some fourth power means of two-term exponential sums, and give exact calculating formulae and asymptotic formula for them.

References

[1]	W. P. Zhang, D. Han, On the sixth power mean of the two-term exponential sums, J. Number Theo., 136 (2014), 403–413. https://doi.org/10.1016/j.jnt.2013.10.022 doi: 10.1016/j.jnt.2013.10.022
[2]	W. P. Zhang, Y. Y. Meng, On the sixth power mean of the two-term exponential sums, Acta Math. Sin., 38 (2022), 510–518. https://doi.org/10.1007/s10114-022-0541-8 doi: 10.1007/s10114-022-0541-8
[3]	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
[4]	W. P. Zhang, H. L. Li, Elementary Number Theory, Shaanxi Normal University Press, Xi'an, 2013.
[5]	H. Zhang, W. P. Zhang, The fourth power mean of two-term exponential sums and its application, Math. Rep., 19 (2017), 75–81.
[6]	T. T. Wang, W. P. Zhang, On the fourth and sixth power mean of mixed exponential sums, Sci. Sin. Math., 38 (2011), 265–270. https://doi.org/10.1360/012010-753 doi: 10.1360/012010-753
[7]	W. P. Zhang, J. Y. Hu, The number of solutions of the diagonal cubic congruence equation $\bmod p$, Math. Rep., 20 (2018), 70–76.
[8]	B. C. Berndt, R. J. Evans, The determination of Gauss sums, Bull. Am. Math. Soc., 5 (1981), 107–128. https://doi.org/10.1090/S0273-0979-1981-14930-2 doi: 10.1090/S0273-0979-1981-14930-2
[9]	Z. Y. Chen, W. P. Zhang, On the fourth-order linear recurrence formula related to classical Gauss sums, Open Math., 15 (2017), 1251–1255. https://doi.org/10.1515/math-2017-0104 doi: 10.1515/math-2017-0104
[10]	L. Chen, On the classical Gauss sums and their some properties, Symmetry, 10 (2018), 625. https://doi.org/10.3390/sym10110625 doi: 10.3390/sym10110625
[11]	W. P. Zhang, X. D. Yuan, On the classical Gauss sums and their some new identities, AIMS Math., 2022 (2022), 5860–5870. https://doi.org/10.3934/math.2022325 doi: 10.3934/math.2022325
[12]	T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. https://doi.org/10.1007/978-1-4757-5579-4
[13]	K. Ireland, M. Rosen, A classical introduction to modern number theory, Springer-Verlag, New York, 1982. https://doi.org/10.1007/978-1-4757-1779-2

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