A certain new Gauss sum and its fourth power mean

Yan Zhao; Wenpeng Zhang; Xingxing Lv; Yan Zhao; Wenpeng Zhang; Xingxing Lv

doi:10.3934/math.2020321

AIMS Mathematics

2020, Volume 5, Issue 5: 5004-5011. doi: 10.3934/math.2020321

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Research article

A certain new Gauss sum and its fourth power mean

1.
College of Information Engineering, Xian University, Xi’an, Shaanxi, P. R. China
2.
School of Mathematics, Northwest University, Xi’an, Shaanxi, P. R. China

Received: 13 April 2020 Accepted: 01 June 2020 Published: 09 June 2020
MSC : 11L03, 11L07

The main purpose of this paper is using the elementary methods and the properties of the Legendre symbol to study the computational problem of the fourth power mean of a certain generalized quadratic Gauss sum, and give two exact calculating formulae for it.
- congruence equation,
- certain generalized quadratic Gauss sums,
- the fourth power mean,
- calculating formula
Citation: Yan Zhao, Wenpeng Zhang, Xingxing Lv. A certain new Gauss sum and its fourth power mean[J]. AIMS Mathematics, 2020, 5(5): 5004-5011. doi: 10.3934/math.2020321

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Abstract

The main purpose of this paper is using the elementary methods and the properties of the Legendre symbol to study the computational problem of the fourth power mean of a certain generalized quadratic Gauss sum, and give two exact calculating formulae for it.

References

[1]	Tom M. Apostol, Introduction to Analytic Number Theory, Springer Science & Business Media, 1976.
[2]	A. Weil, On some exponential sums, P. Natl. Acad. Sci. USA., 34 (1948), 204-207. doi: 10.1073/pnas.34.5.204
[3]	W. P. Zhang, X. Lin, On the fourth power mean of the generalized quadratic Gauss sums, Acta Math. Sin., 34 (2018), 1037-1049. doi: 10.1007/s10114-017-7188-x
[4]	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., 504 (2013), 1-8.
[5]	X. X. Lv, W. P. 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. doi: 10.1007/s10986-017-9366-z
[6]	S. Chern, On the power mean of a sum analogous to the Kloosterman sum, Bull. Math. Soc. Sci. Math. Roumanie, 62 (2019), 77-92.
[7]	W. P. Zhang, Moments of generalized quadratic Gauss sums weighted by L-functions, J. Number Theory, 92 (2002), 304-314. doi: 10.1006/jnth.2001.2715
[8]	H. Zhang, W. P. Zhang, The fourth power mean of two-term exponential sums and its application, Math. Rep., 19 (2017), 75-81.
[9]	H. Di, A Hybrid mean value involving two-term exponential sums and polynomial character sums, Czech. Math. J., 64 (2014), 53-62. doi: 10.1007/s10587-014-0082-0
[10]	W. P. Zhang, On the fourth and sixth power mean of the classical Kloosterman sums, J. Number Theory, 131 (2011), 228-238. doi: 10.1016/j.jnt.2010.08.008
[11]	W. P. Zhang, On the fourth power mean of the general Kloosterman sums, Indian J. Pure Ap. Mat., 35 (2004), 237-242.
[12]	W. P. Zhang, On the fourth power mean of the general Kloosterman sums, J. Number Theory, 169 (2016), 315-326. doi: 10.1016/j.jnt.2016.05.018
[13]	W. P. Zhang, S. M. Shen, A note on the fourth power mean of the generalized Kloosterman sums, J. Number Theory, 174 (2017), 419-426. doi: 10.1016/j.jnt.2016.11.020

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