A Study On The Mean Of Gauss Sums And Two-term Exponential Sums

For a long time,the mean value estimation problems of some well-known sums in analytic number theory,such as Gauss sums,two-term exponential sums and Kloosterman sums and their various generalized sums,have been a hot issue concerned by many scholars,and many famous number theory problems are also closely related to it.In this paper,we study the mean value of several important sums,including the special hybrid power mean of the generalized Gauss sums,the hybrid power mean of the cubic Gauss sums and Kloosterman sums,the fourth power mean of the two-term exponential sums and the mean of the two-term exponential sums and the cubic Gauss sums.At the same time,the divisibility properties of some well-known polynomials such as Chebyshev polynomials,Lucas polynomials and other famous polynomials are also studied.Specifically,the main results of this paper are as follows:1.Regarding the hybrid power mean of Gauss sums and other sums.Firstly,we study the problem of the hybrid power mean of generalized Gauss sums with symmetric form,and obtain a calculating formula of their 2kth power mean under certain conditions.Secondly,the high-order hybrid power mean of the cubic Gauss sums and Kloosterman sums is studied,an interesting linear recursion formula and the corresponding strong asymptotic formula are given respectively.2.Regarding the hybrid power mean of the two-term exponential sums and other sums.This paper studies the fourth power mean of the two-term exponential sums,which extends on previous research.The condition that the power of the first term is odd is improved to the condition of even number,and gives the precise computational formulas respectively for(?)where p?3 mod 4 and p?1 mod 4.These results are helpful to study the mean value problem of its higher power.In addition,the hybrid power mean of two-term exponential sums and the cubic Gauss sums is studied,and some identities and asymptotic formulas are obtained.Meanwhile,a third order linear recursion formula for them is given.3.Regarding the properties and applications of some polynomials.This paper uses the properties of Chebyshev polynomials to study the power sum problem of Fibonacci polynomials and Lucas polynomials.Some new divisible properties of these polynomials are given.At the same time,Girard and Waring formulas and mathematical induction are used to solve the conjecture proposed by Melham.In addition,the recurrence properties of a kind of rational polynomi-als of the classical Gauss sum are also studied,and the corresponding recurrence formula is obtained.Finally,some new identities of Tribonacci numbers are given.