Mixed Mean Studies Of Several Classes Of Exponential Sums

Exponential sum is one of the important contents in the analytic number theory.The researches of exponential sum are closely related to many famous intractable problems in number theory.Therefore,the discussion of exponential sum has the profound theoretical value and the wide application prospects.Not only the exact calculation and the upper bound estimation of exponential sum,but also the hybrid mean value related to exponential sum is very important.Inspired by many scholars's in-depth researches which are related to exponential sum,this thesis is aimed at studying the hybrid mean value involving general Kloosterman sums K(r,l,?,p)and Hardy sums S1(h,p).By using the elementary and analytic methods,we obtain some exact computational formulae and upper bounds under different conditions.Moreover,we also study the hybrid mean value involving general cubic Gauss sums and a special kind of exponential sum.The main achievements contained in this thesis are as follows:1.The hybrid mean value involving general Kloosterman sums K(r,l,?;p)and Hardy sums S1(h,p)in the form of:#12#12 where#12 r,l,h are integers,p is odd prime,A is the character modulo p.Using the properties of the Gauss sums,the orthogonality relation for character sum and the mean value of the Dirichlet L-function,we obtain some interesting results.2.The hybrid mean value involving general cubic Gauss sums and a special kind of exponential sum in the form of#12#12 By using the trigonometric identity,the properties of the Dirichlet character and the classical Gauss sums,we give some exact computational formulae.