On The Mean Value Of Some Exponential Sums

In analytic number theory,Kloosterman sums,Guass sums and Ramanu-jan sums etc.have a close relationship between them.In recent years,many problems have been studied,and got several excellent results.Basing on previous research achievements of the various exponential sums,and using the simplified residue system,and the triangular sum,the orthogo-nality of the Dirichlet character,Ramanujan sums and the methods of related references to study some exponential sums.The main achievements arc listed:1.Let q,m,n,k,s be positive integers with q? 3,(n,q)= 1,for(?)we study the mean value of the above exponential sum,and get a result.2.Let t be a positive integer,k1,k2,…,kt and m1,m2,…,mt,n be positive integers and p be prime,and let q>2 be a positive integer with(m2,q)=…=(mt,q)=(n,q)= 1.Let ?(x)denote m1xk1+m2xk2?…+mtxkt,L = k1?k2+…+kt.for(?)we study the mean value of the above exponential sum when(k1,q)= 1 and k1|k2,k1|k3,…,k1|kt,and for any positive integer n with(n,q)= 1;or(ki,q?(q))= 1,and for any positive integer n with(n,p)= 1,and we obtain two formulas.3.Let p.m,,k,l,n,a be positive integers,and p be a prime,(n,p)= 1,for(?)we study the mcan value of the above exponential sum,and we obtain some results.