Study On The Mean Of The Two-term Exponential Sums And Their Applications

It is well known that the two-term exponential sums are important research fields in analytic number theory,aiming at the upper bound estimation.In this paper,by using the properties of the two-term exponential sums,combined with the character theory and congruence theory,we study the recurrence property of a kind of character sum,the mean value of two-term exponential sums and one kind hybrid power mean of character sums and two-term exponential sums.As applications,we study the power sum problem of Lucas polynomials and its divisible property,and we study the solution of congruence equation.Specifically,the main results of the study are summarized as follows:1.In the second chapter,we study a kind of character sums Ak(h,?1,?2,…,?k;p)and their recurrence properties.For any positive integers k and h,we consider the computation of the character sums,with an odd prime p,Ak(h,?1,?2,…,?k;p)=(?)?1(a1)?2(a2)…?k(ak),where ?i(i=1,2,…,k)are the Dirichlet characters modulo p.Firstly,under some conditions on p and the characters ?i(i=1,2,···,k),we give an exact computational formula for Ak(p)=Ak(3,?2,?2,…,?2;P).Secondly,we obtain that Ak(p)satisfies an interesting third order linear recurrence formula when p?1 mod 6.Finally,combined with the important work of B.C.Berndt and R.J.Evans,we give the third order linear recurrence formula for Ak(p)when p?1 mod 6 and 2 is a cubic residue modulo p.We use the conclusions of analytic number theory such as the properties of Gauss sums,the properties of Dirichlet characters and reduced residue systems modulo p.2.In the third chapter,we study the fourth power mean of the two-term ex-ponential sums.Based on congruence theory,the properties of two-term exponential sums and trigonometric sums,when p is an odd prime number,we give the precise computational formulas respectively for where 5(?)(p-1)and 5|(p-1).3.In the fourth chapter,we study one kind hybrid power mean of three-term character sums and two-term exponential sums.Using the theory of character sums and Gauss sums,when p is an odd prime with(3,p-1)=3,we obtain the exact computational formula for the sum4.In the fifth chapter,we study the power sum problems of Lucas polynomi-als and Fibonacci polynomials and their divisible properties.Using the mathematical induction and the properties of Fibonacci polynomials and Lucas polynomials,we study the divisible properties for the following sums L1(x)L3(x)…L2n+1(x)(?)F2m+12n+1(x),L1(x)L3(x)…L2n+1(x)(?)L2m+12n+1(x),In this chapter,it is actually the further research for Melham's conjecture.5.In the sixth chapter,as the application of the second chapter,using the re-sult of Ak(p)and character theory,when p is a prime number in the form of p?2 mod 3,we give the solution number for the congruence equation x6+y6+z6?0 mod p in Zp3.Using the method of partition,we further study the classification of solutions,and get the exact computational formula for the different solution types.