A Study Of The Mean Value Of Several Types Of Sums

Dirichlet sums,Kloosterman sums,Gauss sums and Dedekind sums play important roles in analytic number theory.Many scholars have studied many important number theory problems by using these sums.Therefore,it is particularly important to study these sums themselves.However,the values of these sums are not regular,so we need to further research the properties of these sums by means of the study of their mean value.Focusing on this idea and based on existing research results,this paper mainly conducts the following studies:1.Let p be an odd prime,A be a Dirichlet character modulo p.With the help of important relationships among Hardy sums and Dedekind sums,we obtain some exact computational formulas or upper bounds for hybrid mean value involving Hardy sums S3(h,p)and general Kloosterman sums K(r,l,?;p)in the form of (?) where (?)We find that for most of other forms of Hardy sums,we can't have the similar discussion.2.We use two different methods to discuss some different sums which contain Dirichlet characters of some certain rational polynomials,and give some exact formulas.Let X be any primitive even character modulo q.When q is an odd number or an odd square-full number,we consider the sum as follows:(?) and obtain exact computational formulas or upper bounds respectively.Also,when q is an odd square-full number,we derive a new identity in the form of (?)When q=p is an odd prime,? is any non-principal character modulo p,we obtain a new identity about (?).