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JP Journal of Algebra, Number Theory and Applications

JP Journal of Algebra, Number Theory and Applications
Volume 64, Issue 4, Pages 433 - 450 (July 2025)
http://dx.doi.org/10.17654/0972555525023

THE HYBRID POWER MOMENT OF CHARACTER SUMS AND TWO-TERM EXPONENTIAL SUMS

Xuan Wang

Abstract:

This paper focuses on the investigation of the hybrid power mean concerning some special character sums of polynomials and the two-term exponential sums by utilizing analytical approaches and the nature of Gauss sums to give two sharp asymptotic formulae for them.

Keywords and phrases:

character sums of polynomials, two-term exponential sums, asymptotic formula.

Received: March 30, 2025; Revised: May 6, 2025; Accepted: May 19, 2025; Published: June 11, 2025

How to cite this article: Xuan Wang, The hybrid power moment of character sums and two-term exponential sums, JP Journal of Algebra, Number Theory and Applications 64(4) (2025), 433-450. https://doi.org/10.17654/0972555525023

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1]D. Han, A hybrid mean value involving two-term exponential sums and polynomial character sums, Czechoslovak Math. J 64 (2014), 53-62.

[2]X. Y. Du, The hybrid power mean of two-term exponential sums and character sums, Acta Math. Sin. (Chin. Ser.) 59 (2016), 309-316.

[3]W. P. Zhang and Y. Yi, On Dirichlet characters of polynomials, Bull. London Math. Soc. 34 (2002), 469-473.

[4]D. A. Burgess, On Dirichlet characters of polynomials, Proceedings of the London Mathematical Society 13 (1963), 537-548.

[5]Y. K. Ma and W. P. Zhang, On the two-term exponential sums and character sums of polynomials, Open Mathematics 17 (2019), 1239-1248.

[6]T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.

[7]K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1982.

[8]X. X. Lv and W. P. Zhang, On the character sum of polynomials and the two-term exponential sums, Acta Math. Sin. Engl. Ser. 36 (2020), 196-206.

[9]X. X. Lv and W. P. Zhang, A new hybrid power mean involving the generalized quadratic Gauss sums and sums analogous to Kloosterman sums, Lithuanian Math. J. 57 (2017), 359-366.

[10]Z. F. Xu, T. P. Zhang and W. P. Zhang, On the mean value of the two-term exponential sums with Dirichlet characters, J. Number Theory 123 (2007), 352-362.

[11]W. P. Zhang, The fourth and sixth power mean of the classical Kloosterman sums, J. Number Theory 131 (2011), 228-238.

[12]H. Zhang and W. P. Zhang, The fourth power mean of two-term exponential sums and its application, Mathematical Reports 19 (2017), 75-81.

[13]X. Wang, L. Wang and G. H. Chen, The fourth power mean of the generalized quadratic Gauss sums associated with some Dirichlet characters, AIMS Mathematics 9 (2024), 17774-17783.

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