指数和的渐近公式

AN ASYMPTOTIC FORMULAR OF TRIGONOMETRICAL SUMS

设A(α ,x) = x≤ne axkq 满足 (a ,q) =1,n为一个正整数 .1≤d≤D <q2 ,这里d ,D是正整数 .记S(q ,a) = qa =1e amkq ,E = nm =11km1k-1.在该文中证明了如下的渐近公式A(α ,x) =q-1d-1S(q ,adk)E+Oq12 +(q,d) .

Suppose A(α,x)=x≤n ea x kq satisfying (a,q)=1, n is a positive integer.1≤d≤D<q2,where d and D are positive integers. Let S(q,a)=qa=1eam kq, E=nm=11km 1k-1 .In this paper we proved the following asymptotic formular: A(α,x)=q -1 d -1 S(q,ad k)E+O(q 12+ (q,d)).