Xu Xiaoling, Zhang Jiafan and Zhang Wenpeng

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 1, Pages 112–124

DOI: 10.7546/nntdm.2021.27.1.112-124

**Full paper (PDF, 201 Kb)**

## Details

### Authors and affiliations

Xu Xiaoling

*School of Data Science and Engineering, Xi’an Innovation College of Yan’an University
Xi’an, Shaanxi, P. R. China*

Zhang Jiafan

*School of Mathematics, Northwest University
Xi’an, Shaanxi, P. R. China*

Zhang Wenpeng

*School of Mathematics, Northwest University
Xi’an, Shaanxi, P. R. China*

### Abstract

The main purpose of this paper is using the properties of the classical Gauss sums and the analytic methods to study the computational problem of one kind of hybrid power mean involving the character sums of polynomials with *k* variables and the two-term exponential sums, and give an identity and asymptotic formula for it.

### Keywords

- Character sums of polynomials with
*k*variables - Two-term exponential sums
- Hybrid power mean
- Analytic method
- Identity
- Asymptotic formula

### 2010 Mathematics Subject Classification

- 11L03
- 11L40

### References

- Apostol, T. M. (1976). Introduction to Analytic Number Theory, Springer-Verlag, New York.
- Bourgain, J. M., Garaev, Z., Konyagin, S. V., & Shparlinski, I. E. (2012). On the hidden shifted power problem. SIAM J. Comput., 41, 1524–1557.
- Chern, S. (2019). On the power mean of a sum analogous to the Kloosterman sum. Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie, 62, 77–92.
- Han, D. (2014). A Hybrid mean value involving two-term exponential sums and polynomial character sums. Czechoslovak Mathematical Journal, 64, 53–62.
- Hou, Y. W., & Zhang, W. P. (2018). One kind high dimensional Kloosterman sums and its upper bound estimate. Journal of Shaanxi Normal University (Natural Science Edition), 46,28–31.
- Liu, Y. Y., & Zhang, W. P. (2017). The linear recurrence formula of the hybrid power mean involving the cubic Gauss sums and two-term exponential sums. Journal of Shaanxi Normal University (Natural Science Edition), 45, 14–17.
- Lv, X. X., & Zhang, W. P. (2017). A new hybrid power mean involving the generalized quadratic Gauss sums and sums analogous to Kloosterman sums. Lithuanian Mathematical Journal, 57, 359–366.
- Lv, X. X., & Zhang, W. P. (2020). On the character sum of polynomials and the two-term exponential sums. Acta Mathematica Sinica, English Series, 36, 196–206.
- Pan, C. D., & Pan, C. B. (1992). Goldbach Conjecture, Science Press, Beijing.
- Smith, R. A. (1979). On
*n*-dimensional Kloosterman sums. Journal of Number Theory, 11,324–343. - Wang, J. Z., & Ma, Y. K. (2017). The hybrid power mean of the
*k*-th Gauss sums and Kloosterman sums. Journal of Shaanxi Normal University (Natural Science Edition), 45,5–7. - Ye, Y. (2000). Estimation of exponential sums of polynomials of higher degrees II. Acta Arithmetica, 93, 221–235.
- Zhang, H., & Zhang, W. P. (2017). The fourth power mean of two-term exponential sums and its application. Mathematical Reports, 19, 75–81.
- Zhang, W. P. (2011). On the fourth and sixth power mean of the classical Kloosterman sums. Journal of Number Theory, 131, 228–238.
- Zhang, W. P. (2016). On the fourth power mean of the general Kloosterman sums. Journal of Number Theory, 169, 315–326.
- Zhang, W. P., & Han, D. (2016). A new identity involving the classical Kloosterman sums and 2-dimensional Kloosterman sums.International Journal of Number Theory, 12,111–119.
- Zhang, W. P., & Li, X. X. (2017). The fourth power mean of the general 2-dimensional Kloosterman sums mod p. Acta Mathematica Sinica, English Series, 33, 861–867.
- Zhang, W. P.,& Lv, X. X. (2019). The fourth power mean of the general 3-dimensional Kloosterman sums mod p. Acta Mathematica Sinica, English Series, 35, 369–377.
- Zhang, W. P., & Shen, S. M. (2017). A note on the fourth power mean of the generalized Kloosterman sums. Journal of Number Theory, 174, 419–426.
- Zhang, W. P., & Yao, W. L. (2004). A note on the Dirichlet characters of polynomials. Acta Arithmetica, 115, 225–229.
- Zhang, W. P., & Yi, Y. (2002). On Dirichlet characters of polynomials. Bulletin of the London Mathematical Society., 34, 469–473

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## Cite this paper

Xiaoling, X., Jiafan, Z., & Wenpeng, Z. (2021). The character sum of polynomials with *k* variables and two-term exponential sums. *Notes on Number Theory and Discrete Mathematics*, 27(1), 112-124, DOI: 10.7546/nntdm.2021.27.1.112-124.