@article{Guo2022, 
author = {Wenjia Guo and Xiaoge Liu and Tianping Zhang},
title = {Dirichlet characters of the rational polynomials},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {3},
pages = {3494-3508},
keywords = {Dirichlet characters, Gauss sums, rational polynomials, fourth power mean, identity},
url = {https://www.sciopen.com/article/10.3934/math.2022194},
doi = {10.3934/math.2022194},
abstract = {Denote by    χ a Dirichlet character modulo    q  ≥  3, and        a    ¯   means    a  ⋅      a    ¯    ≡  1  mod  q. In this paper, we study Dirichlet characters of the rational polynomials in the form               ∑              a        =        1                    q              ′    χ  (  m  a  +      a    ¯    )  ,where              ∑              a        =        1                    q              ′   denotes the summation over all    1  ≤  a  ≤  q with    (  a  ,  q  )  =  1. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity [6] under a more relaxed situation. We also derive some new identities for the fourth power mean of it by adding some new ingredients.}
}