On the classical Gauss sums and their some new identities

Wenpeng Zhang; Xiaodan Yuan; Wenpeng Zhang; Xiaodan Yuan

doi:10.3934/math.2022325

AIMS Mathematics

2022, Volume 7, Issue 4: 5860-5870. doi: 10.3934/math.2022325

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On the classical Gauss sums and their some new identities

Wenpeng Zhang ,
Xiaodan Yuan ^,

School of Mathematics, Northwest University, Xi'an, Shaanxi, China

Received: 23 October 2021 Revised: 06 January 2022 Accepted: 07 January 2022 Published: 12 January 2022
MSC : 11L10, 11L40

In this paper, we use the analytic methods and the properties of the classical Gauss sums to study the calculating problems of some Gauss sums involving the character of order $ 12 $ modulo an odd prime $ p $, and obtain several new and interesting identities for them.
- the character of order $ 12 $,
- the classical Gauss sums,
- analytic methods,
- identity,
- calculating formula
Citation: Wenpeng Zhang, Xiaodan Yuan. On the classical Gauss sums and their some new identities[J]. AIMS Mathematics, 2022, 7(4): 5860-5870. doi: 10.3934/math.2022325

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Abstract

In this paper, we use the analytic methods and the properties of the classical Gauss sums to study the calculating problems of some Gauss sums involving the character of order $ 12 $ modulo an odd prime $ p $, and obtain several new and interesting identities for them.

References

[1]	T. M. Apostol, Introduction to analytic number theory, New York: Springer, 1976.
[2]	K. Ireland, M. Rosen, A classical introduction to modern number theory, New York: Springer, 1982. https://doi.org/10.1007/978-1-4757-1779-2
[3]	C. D. Pan, C. B. Pan, Goldbach conjecture, Beijing: Science Press, 1992.
[4]	W. P. Zhang, J. Y. Hu, The number of solutions of the diagonal cubic congruence equation $\bmod p$, Math. Reports, 20 (2018), 73–80.
[5]	B. C. Berndt, R. J. Evans, The determination of Gauss sums, Bull. Amer. Math. Soc., 5 (1981), 107–128. https://doi.org/10.1090/S0273-0979-1981-14930-2 doi: 10.1090/S0273-0979-1981-14930-2
[6]	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
[7]	W. P. Zhang, H. L. Li, Elementary number theory (Chinese), Xi'an: Shaanxi Normal University Press, 2013.
[8]	L. Chen, On classical Gauss sums and some of their properties, Symmetry, 10 (2018), 1–6. https://doi.org/10.3390/sym10110625 doi: 10.3390/sym10110625
[9]	L. Chen, Z. Y. Chen, Some new hybrid power mean formulae of trigonometric sums, Adv. Differ. Equ., 2020 (2020), 1–9. https://doi.org/10.1186/s13662-020-02660-7 doi: 10.1186/s13662-020-02660-7
[10]	L. Chen, J. Y. Hu, A linear recurrence formula involving cubic Gauss sums and Kloosterman sums, Acta Math. Sinica (Chin. Ser.), 61 (2018), 67–72.
[11]	S. M. Shen, W. P. Zhang, On the quartic Gauss sums and their recurrence property, Adv. Differ. Equ., 2017 (2017), 1–9. https://doi.org/10.1186/s13662-017-1097-2 doi: 10.1186/s13662-017-1097-2
[12]	S. Chowla, J. Cowles, M. Cowles, On the number of zeros of diagonal cubic forms, J. Number Theory, 9 (1977), 502–506. https://doi.org/10.1016/0022-314X(77)90010-5 doi: 10.1016/0022-314X(77)90010-5
[13]	W. P. Zhang, The fourth power mean of the generalized quartic Gauss sums, J. Shaanxi Normal Univ. Nat. Sci. Ed., 49 (2021), 1–5.
[14]	T. T. Wang, On the fourth power mean of the generalized two-term exponential sums, J. Shaanxi Normal Univ. Nat. Sci. Ed., 49 (2021), 6–12.
[15]	J. Greene, D. Stanton, The triplication formula for Gauss sums, Aequationes Math., 30 (1986), 143–141. https://doi.org/10.1007/BF02189920 doi: 10.1007/BF02189920
[16]	W. P. Zhang, A. Samad, Z. Y. Chen, New identities dealing with Gauss sums, Symmetry, 12 (2020), 1–7. https://doi.org/10.3390/sym12091416 doi: 10.3390/sym12091416
[17]	W. Duke, H. Iwaniec, A relation between cubic exponential and Kloosterman sums, Contemp. Math., 143 (1993), 255–258.
[18]	H. Davenport, H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen F$\ddot{a}$llen, J. Reine Angew. Math., 172 (1935), 151–182. https://doi.org/10.1515/crll.1935.172.151 doi: 10.1515/crll.1935.172.151

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