Study On Kloosterman Sums And Their Applicaltions Based On The Method Of Character Sums

In 1912,the Kloosterman sums first appeared in a paper of Henri Poincareon modular forms.In 1926.,H.D.Kloosterman proposed a kind of sums on complex field when he studied the problem of representation of large positive integers in the form of sum-of-squares by means introduced by Hardy and Littlewood.In analytic number theory,this kind of sums are very important and have great research value.However,we note that the index of both a and ??? in the classic Kloosterman are 1.In fact,when we deal with the relative problems,it always refer to certain sums either with the Dirichlet character,or the index of a and ??? are not all 1.Therefore,the classic Kloosterman sums should be generalized.and for further study we define two kinds of new Kloosterman sums.They are?r,s?-Kloosterman sums K?r,s,m,n;q?and generalized Kloosterman sums.In general,based on the methods of characters sums and the properties of Gauss sums,and combining the congruence theory,we study the high-th power mean of these new kind Kloosterman sums.And we intend to give an exact computational formula or asymptotic formula for one kind hybrid power mean of three-term character sums and two-term exponential sums.As an application,we give calculate the solutions of one kind congruence equation.Finally,we obtain an identity involving the integral of the first kind Chebyshev polynomials and prove a conjecture proposed by Chen Yonggao.?1?Using the properties of congruence,we give the computational formula for the fourth power mean of the?r,s?-Kloosterman surms K?r,s,m,n;q?when parameters q,r,and s in?r,s?-Kloostirman sums assume special values.Moreover,we shall combine B.J.Birch's important work,S.Chowla,J.Cowles,M.Cowles's works and W.Duke,H.Iwaniec's results to study 2k-th power mean of general Kloosterman sums.In addition,using the the properties of trigonometric sum and Gauss sums,we obtain some some exact computational formulae for 2k-th power mean of general Kloosterman sums with small integer k and a special non-principal character ? mod p.?2?Based on the properties of character sum and congruent theory,we give an exact computational formula or asymptotic forImula for one kind hybrid power mean of three-term character sums and two-term exponential sums with k = 3,h = 2.?3?As an application of exponential sums,we mainly use the proper-ties of character sums and the methods of mathematical analysis to give a formula for the number of solutions to the cubic congruence equation x₁³+x₂³+ x₃³+x₄³?c mod p when c?0 mod p.And as a corollary,we can also derive the formula for the four-th mean value of two-term exponential sums.?4?Chebyshev polynomials is one kind of orthogonal polynomials.As we know that many of their properties play important roles in numerical analysis,engineering mathematics,financial mathematics,nonlinear math-ematical physical equation.We obtain an identity involving convolution integral of the first kind Chebyshev polynomials.The methods we mainly use are the algebraic method and the properties of first kind Chebyshev polynomials.?5?Chen Yonggao has proposed two conjectures about Levine-O'Sullivan sequence {q_n}.Yang Shichun and Jiang Zigco have studied these conjectures and proved the second one.But the first conjecture has not been proved yet.And in this paper,We continue this research and proved the first conjecture.That is we will prove q_n =?^?-3n^?+o?n^??Thatis we will pr1ve qn=??-3n?+o?nw?.The methods we mainly use are mathematical induction and mathematical analysis.