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Keywords:
mixed exponential sum; mean value; Dirichlet character; general Gauss sum; computational formula
Summary:
The main purpose of the paper is to study, using the analytic method and the property of the Ramanujan's sum, the computational problem of the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum. For integers $m$, $ n$, $ k$, $ q$, with $k\geq {1}$ and $q\geq {3}$, and Dirichlet characters $\chi $, $\bar {\chi }$ modulo $q$ we define a mixed exponential sum $$ C(m,n;k;\chi ;\bar {\chi };q)= \sum \limits _{a=1}^{q}{\mkern -4mu\vrule width0pt height1em}' \chi (a)G_{k}(a,\bar {\chi })e \Big (\frac {ma^{k}+n\overline {a^{k}}}{q}\Big ), $$ with Dirichlet character $\chi $ and general Gauss sum $G_{k}(a,\bar {\chi })$ as coefficient, where $\sum \nolimits '$ denotes the summation over all $a$ such that $(a,q)=1$, $a\bar {a}\equiv {1}\mod {q}$ and $e(y)={\rm e}^{2\pi {\rm i} y}$. We mean value of $$ \sum _{m}\sum _{\chi }\sum _{\bar {\chi }}|C(m,n;k;\chi ;\bar {\chi };q)|^{4}, $$ and give an exact computational formula for it.
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