Proof Of Uniform Distribution Of Chebyshev Sequ-Ences And Maximal Curve Sequences

Analytical number theory is a branch of number theory that uses analytical methods as research tools,and exponential sums and character sums are widely used as important research tools in analytical number theory.For example,the proof of uniform distribution of sequences can be converted to solving the upper bound of exponential sums;The number of solutions of the congruence equation can be converted to the expression of exponential sums upper bounds,and the exact expression of the number of solutions can be given and so on.In this thesis,the properties and applications of exponential sum and characteristic sum are studied,that is,the Chebyshev polynomial sequence and maximal curve sequence are constructed,and the upper bound of the discrepancy of the constructed sequence is calculated by using the properties of exponential sums,so as to prove the uniform distribution property of the sequences.The number of solutions of the constructed congruence equations is estimated by using exponential sums properties,and the character sums analogous to high dimensional Kloosterman sums is calculated.The specific research contents and results are as follows:(1)The Chebyshev polynomial sequence is constructed by polynomial iteration,and an upper bound of the discrepancy is obtained by using exponential sums properties,so as to prove the uniform distribution property of the sequence.(2)A maximal curve sequence is constructed by extracting rational points on the maximal curve and combining with the pseudorandom number generator.The upper bound of the discrepancy of the maximal curve is obtained by using the property of exponential sums,which showed the uniform distribution property of the sequence.(3)A new congruence equation λx1-ωx2≡x3(mod p)is constructed on the basis of previous research and the number of solutions is estimated by using the properties of exponential sums and some inequalities.(4)The character sums of a class of high dimensional Kloosterman sums is calculated by using the classical Gauss sums properties.And some interesting identities are given.