Research On The Mean Of Exponential Sums And Their Applications

The mean value of exponential sums has always been an important part of number theory,which included the research of Gauss sums,character sums,Kloosterman sums.There are also close connections between them in a long history.Many scholars have conducted in-depth discussions on related issues and have achieved progressive results.Based on the interests in the above problems,first of all,this paper uses the properties of the character sums and the trigonometric identity to study the mean value of the important sums,including Gauss sums,two-term exponential sums,Kloosterman sums,and their generalized sums.Secondly,we studied a problem related to the number of solutions to one congruence equation.Finally,we researched the properties and applications of some famous polynomials.Specifically,the main results of this paper are as follows :1.Study the mean value related to Gauss sums.Firstly,we introduce a new type of generalized quadratic Gauss sums,which give an accurate calculation formula about its fourth-order mean under certain circumstances.Moreover,we also get the recurrence formulas for one kind hybrid power mean involving two-term exponential sum and Gauss sum depending on the congruence of the modulo.2.Completely solve the computational problem of the fourth power mean of generalized Kloosterman sums.On the basis of previous results,according to the orthogonality of character,we use the properties of Gauss sum and the reduced residue system to study in detail the mean value of the generalized Kloosterman sum.For the non-primitive character of composite numbers ,an accurate calculation formula about the fourth power mean of generalized Kloosterman sums is given.3.Research on the solution of one congruence equation.Using the properties of trigonometric sum identities and Gauss sum to study the calculation problem of one congruence equation with odd prime numbers,and give the recurrence formula for the number of solutions of the congruence equation.4.Study the properties and applications of some polynomials.First of all,relied on the definition and properties of Chebyshev polynomials,the power sum problem of sin and cos is been solved,and giving some regular calculation formulas.Secondly,we also obtained an interesting identity about the convolution sum of Legendre polynomials by introduced a new nonlinear recursive sequence,then the general formula for calculating the convolution sum of Legendre polynomials when the number of terms is odd is obtained.Finally,we prove some new congruences related to Bernoulli polynomials formula which take advantage of the properties of Bernoulli polynomials and Euler polynomials.The relationship between these two polynomials is established for the first time,...