Mixed Means Study On Kloosterman-Type Exponential Sums

During the process of researching,the exponential sums of Kloosterman types are one of the important contents in Analytic Number Theory.Up to now,it has a long history and been widely applied in many aspects.So,either studying individual properties or considering the relevant mean value about the exponential sums of Kloosterman types,all of it has profound research significance.Based on the importance of research which is involving this hybrid mean value,by picking up the appropriate Kloosterman sums,many scholars try to explore the fact whether there is cancellation or not during the research of mean value.Inspired by this,we will discuss the hybrid mean value about the exponential sums of Kloosterman types in this thesis.Meanwhile,we will give some asymptotic formulae or upper bounds about this hybrid mean value in this paper.In this thesis,the specific content of the study contains the following aspects:1.Using the properties of Dirichlet character and Dirichlet L-function,we generalizes the form of hybrid mean value between Dedekind sum S1(h,p)and classical Kloosterman sum K(n1,1,?;p),where the form is The quadratic-hybrid mean value which studied in this paper is relevant to generalized Kloosterman sum K(n1,nk,?;p)and Hardy sum S3(h,p).The specific form about it is as follows:where p is an odd prime,n1,nk,h are integers,? is the Dirichlet characters modulo p,and2.Using the properties of Dirichlet characters and Gauss sums,we will study the hybrid mean value involving a sum analogous to Kloosterman sum and the quartic exponential sum.The form of hybrid mean value is as follows:where p is an odd prime,s is an integer,? is the non-primary Dirichlet character modulo p.According to the relationship between integers s and p,we will give the asymptotic formula or upper bound about this hybrid mean value.