Number Of Classic And Arithmetic Properties

The study on the properties of the mean value problems of some famous summations is the kernel of the research in number theory. In the arithmetic function, character sums, Dedekind sums, Kloosterman sums and Gauss sums have the glorious history, the extremely rich content, and the close relation. In recent years, the domestic and foreign many scholars have conducted the thorough research to these questions, and obtained some important valuable results on theory. This plays an irreplaceable and important role in the field of number theory.Based on our interests in the above problems, the thesis is concerned with the researches on the arithmetic properties of the classical sums in analytic theory and the identities involving the Fibonacci numbers and Lucas numbers, synthetically using the elementary and analytic methods. In addition, the thesis also includes researches on the Diophantine equation and its integer solutions and the properties of the mean value of a new additive function and Smarandache sequences. Specially, the main achievements contained in this dissertation are as follows:1. Researches on the hybrid mean value related to the Dedekind sums and Kloosterman sums. We use the properties of character sums and the analytic method to study a hybrid mean value problem related to the Dedekind sums and Kloosterman sums, and give some interesting mean value formulas and identities for it.2. Researches on the identity involving the Gauss sums and the general Kloosterman sums. We use the properties of Gauss sums and the analytic method to study the relationships between the Gauss sums and the general Kloosterman sums, and give several interesting identities for them.3. Researches on a new kind of polynomials and their power sums. we use the elementary methods to study the sum of powers of this polynomials, and give several interesting identities. Finally, we deduce some identities involving the sums of powers of Fibonacci numbers and Lucas numbers.4. Researches on a Diophantine equation and its integer solutions. We use the elementary method and the divisible properties of the integers to study the Diophantine equation xy+yz+zx= 0 and solve it completely.5. Researches on the properties of the mean value of a new additive function and Smarandache sequences. We introduce a new additive function F(n), then study the mean value properties of F(n) in some special sequences. We use the elementary and analytic methods and give two mean value formulas of F(n) in Smarandache divisor product sequences{Pd(n)} and{qd(n)}.