@article{Xu2025, 
author = {Zhefeng Xu and Xiaoying Liu and Luyao Chen},
title = {Hybrid mean value involving some two-term exponential sums and fourth Gauss sums},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {3},
pages = {1510-1522},
keywords = {two-term exponential sums, fourth Gauss sums, mean value, asymptotic formulas, identities},
url = {https://www.sciopen.com/article/10.3934/era.2025071},
doi = {10.3934/era.2025071},
abstract = {Let    q  ≥  3 be a positive integer. For any integers    m,    n,    k,    h, the two-term exponential sums    C  (  m  ,  n  ,  k  ,  h  ;  q  ) is defined as    C  (  m  ,  n  ,  k  ,  h  ;  q  )  =      ∑          a      =      1              q        e      (                  m                  a          k                +        n                  a          h                    q        )  , where    k  >  h  ≥  2. The main purpose of this paper is to use analytic methods and the properties of classical Gauss sums to study the mean value involving two-term exponential sums and fourth Gauss sums, and to provide some asymptotic formulas and identities. Previously, only the case of    h  =  1 had been studied.}
}