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.
展开▼
机译：摘要：本文的主要目的是使用分析方法和拉曼努扬和的性质研究具有狄利克雷特征和一般高斯和的混合指数和的平均值的计算问题。对于整数$ m $，$ n $，$ k $，$ q $，以及$ k \ geq {1} $和$ q \ geq {3} $，以及Dirichlet字符$ \ chi $，$ \ bar {\ chi} $模$ q $我们定义了一个混合指数和$$ $$ 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），$以Dirichlet字符$ \ chi $和一般高斯和$ G_ {k}（a，\ bar {\ chi}）$作为系数，其中$ \ sum \ nolimits' $表示所有$ a $的总和，使得$（a，q）= 1 $，$ a \ bar {a} \ equiv {1} \ mod {q} $和$ e（y）= {\ rm e } ^ {2 \ pi {\ rm i} y} $。我们的平均值为$$ \ sum _ {m} \ sum _ {\ chi} \ sum _ {\ bar {\ chi}} | C（m，n; k; \ chi; \ bar {\ chi}; q ）| ^ {4}，$$并为其提供精确的计算公式。
展开▼
