Mean value of mixed exponential sums

Abstract:

For integers $q$, $m$, $n$, $k$ with $q,k\geq 1$, and Dirichlet character $\chi \bmod q$, we define a mixed exponential sum \[ C(m,n,k,\chi ;q):= {\sum }’_{a=1}^q\chi (a)\mathrm {e}\left (\frac {ma^k+na}{q}\right ), \] where $\displaystyle \mathrm {e}(y)=\mathrm {e}^{2\pi iy}$, and $\sum ’_{a}$ denotes the summation over all $a$ with $(a,q)=1$. The main purpose of this paper is to study the mean value of \[ \sum _{\chi \bmod q}{\sum }’_{m=1}^q\left |C(m,n,k,\chi ;q)\right |^4, \] and to give a related identity on the mean value of the general Kloosterman sum \[ K(m,n,\chi ;q):={\sum }’_{a=1}^q\chi (a)\mathrm {e}\left (\frac {ma +n\overline {a}}{q}\right ), \] where $a\overline {a} \equiv 1 \bmod q$.