Research On The Mean Of Exponential Sums And Character Sums

The exponential sums and character sums are important research object in analytic number theory,and many famous number theory problems are closely related to it.Estimation of the mean value of exponential sums and character sums has important theoretical meaning and application value.This paper mainly study the arithmetical properties of some well-known summmations,including hyper-Kloosterman sums,two-term exponential sums,Gauss sums and character sums of polynomials.Main results of this paper are as follows.1.Study the relations between the exponential sums and the number of solutions of the congruence equations.Specifically,study the relations between the fourth power mean of the generalized three-dimensional Kloosterman sums and the number of solutions of the congruence equations,and the relations between the sixth power mean of the generalized quadratic two-term exponential sums and the number of solutions of the congruence equations.Obtain the asymptotic formulas for the fourth power mean of the generalized three-dimensional Kloosterman sums,and the calculating formula for the sixth power mean of the generalized quadratic two-term exponential sums.2.Study the hybrid power mean of generalized 6)-th Gauss sums and the sums analogous to generalized Kloosterman sums.This paper study the computational problems for the hybrid power mean of generalized quadratic Gauss sums and the sums analogous to generalized Kloosterman sums involving character,and obtain the exact values or the calculating formulae under different restrictions,which reveal the relations between the generalized quadratic Gauss sums and the sums analogous to generalized Kloosterman sums.3.Obtain some results of the hybrid power mean of 6)-th two-term exponential sums and character sums.Firstly,study the hybrid power mean of3-th two-term exponential sum and character sums of polynomials,and give the exact values or asymptotic formulas.Secondly,study the hybrid power mean of 6)-th two-term exponential sums and character sums of polynomials,and obtain the exact calculation formulas for it under some conditions.Finally,discuss the hybrid power mean of two-term exponential sums and the sums analogous to Kloosterman sums of polynomials,and get some exact values or estimation formulae,which reveal the relation between two-term exponential sums and character sums.4.Study the mixed mean of 6)-th Gauss sums and two-term exponential sums.Firstly,give the recursion formulas of the hybrid power mean of the cubic Gauss sums and 6)-th two-term exponential sums and the recursion formulas of the hybrid power mean of the cubic Gauss sums and generalized6)-th two-term exponential sums.Secondly,we give the recursion formulas of the hybrid power mean of the reciprocal of the quartic Gauss sums and arithmetical function 1)(8)).When (8))is a special function,we give the initial value.