Estimation Of Non-linear Exponential Sums

It is an important problem in analytic number theory to study the exponential sums,which is crucial in many problems in number theory.In this thesis,we consider the following exponential sums We will prove the upper bounds of Sβ(k,x).Firstly,we use the combination of the analytic methods given by Ren[14]and the large value technique presented by Heath-Brown([4])to prove a mean value estimates of Dirichlet series(Theorem 1.2).Secondly,we use Heath-Brown’s identity,Perron’s formula and the mean value estimates of Dirichlet series in Theorem 1.2 to prove our main theorem(Theorem 1.1).Thirdly,as an application of Theorem 1.1,we consider an equi-distribution problem and improve the error term of this question.