Research In Number Theory And Arithmetic Properties

Research on the properties of the mean value problems of some arithmetic function has always been imporant issue in number theory,particulary in Ana-lytic number theory.In the arithmetic function,exponent sums, character sums, Dedekind sums, Kloosterman sums and Gauss sums have the glorious history, the rich content, and the close relation. In recent years, many scholars have conducted the thorough research to these questions, and obtained many results on theory. This plays an important role in the field of number theory.Based on our interests in the above problems, the dissertationis is concerned with the researches on the arithmetic properties of the Hardy sums, Klooster-man sums and Generalized k-th Gauss sums in analytic theory. In addition, the dissertationis researches on the mean value of D. H. Lehmer problem in short interval using the the Fourier expansion for character sums and the estimates for exponential sums.Furthermore, the thesis includes researching on the Pseu-dorandom binary sequence constructed by D. H. Lehmer problem. the thesis also includes researches on the Diophantine equation. Specially,the dissertation gets the following results:1. Researches on the hybrid meanvalue related to certain Hardy sums and Kloosterman sums. We use the mean value formula of Dirichlet L-functions and the analytic method to study a hybrid mean value problem related to cer-tain Hardy sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.2. Researches on the generalized k-th Gauss sums. We use the analytic methods and the properties of the classical Gauss sums to study the computa-tional problem of the2k-th power mean value of the generalized quadratic Gauss sums, and to give an exact computational formula for it.3. Researches on the mean value of D.H. Lehmer problem in short inter-val.We use the the Fourier expansion for character sums and the estimates for exponential sums, and some asymptotic formula is obtained.4. Researches on the Pseudorandom binary sequence constructed by D. H. Lehmer problem. A promotion of D. H. Lehmer problem is given. Thus a new form of pseudorandom sequence is generated. We proves that (en) is a good pseudorandom sequence by using the estimate for exponential sums.5. Researches on the diophantine equation (an-1)(bn-1)=x2. We give all the positive integer solutions (x, n) of the equation.