广义Kloosterman和的四次均值

Fourth power mean of the general Kloosterman sum

设q>2为整数,χ为模q的Dirichlet特征,k为正整数.对任意整数m与n,定义K(m,n,k,χ;q)=∑′qa=1χ(a)e(mak+n/q),其中∑′表示对与q互素的整数a求和,e(y)=exp(2πi y),是a关于模q的乘法逆,满足1≤≤q以及a≡1(modq).本文研究K(m,n,k,χ;q)的四次均值,并给出一些恒等式.

Let q>2 be an integer,and letχbe a Dirichlet character modulo q.Let kbe a positive integer.For arbitrary integers mand n,define K(m,n,k,χ;q)=∑′qa=1χ(a)e(mak+n/q) where ∑′ denotes the summation over all with(a,q)=1,e(y)=exp(2πi y),and ais the inverse of amodulo qsuch that 1≤≤qand a≡1(mod q).The fourth power mean of K(m,n,k,χ;q)was studied,and some identities were given.