Research On The Properties Of Several Famous Functions In Number Theory

The main purpose of this dissertation is to study the arithmetical prop-erties of some famous sum functions, such as generalized Kloosterman sums, Dedekind sums, Gauss sums, two-term character sums and Dirichlet charac-ter of polynomials in analytic number theory. Using the analytic methods to study the computational problems of mean value and hybrid mean value of these functions. Also, using the generating function methods and some com-binatorial techniques to study the Apostol-Bernoulli polynomials, Apostol-Euler polynomials, (p, q)-Fibonacci polynomials and (p, q)-Lucas polynomials in combinatorial number theory, obtained a series new and interesting combi-natorial identities as well as the close formulae of sums of any positive powers for the (p, q)-Fibonacci polynomials and (p,q)-Lucas polynomials. The main achievements contained in this dissertation are as follows:1. Making use of the analytic methods and the properties of Gauss sums to study the hybrid mean value of one kind generalized Kloosterman sums and Dirichlet character of polynomials. An interesting asymptotic formula of them are presented.2. According to the analytic methods and the properties of character sums to study the computational problem of the hybrid mean value involving one kind generalized Kloosterman sums and Dedekind sums. A new identity is proved.3. Using the properties of Gauss sums and the estimate for character sums to study the mean value problem of the two-term character sums. Sev-eral interesting identities for them are obtained.4. Combining the generating function methods with some combinato-rial techniques to establish some new recurrence formulae for the Apostol- Bernoulli polynomials as well as the mixed recurrence formulae for the Apostol-Bernoulli and Apostol-Euler polynomials. It turns out that some known results are obtained as special cases.5. Investigating the properties for the promotion form of classical Fi-bonacci and Lucas polynomials, i.e. (p,q)-Fibonacci and (p, q)-Lucas poly-nomials. By applying some elementary methods and techniques, some com-binatorial identities for the (p,q)-Fibonacci and (p,q)-Lucas polynomials are established while the close formulae of sums of any positive powers for the (p, q)-Fibonacci and (p, q)-Lucas polynomials are obtained.