Dirichlet L-functions?Gauss Sums And Its Applications

The main purpose of this dissertation is to study the mean value problems of some famous summations by using the properties of Dirichlet L-functions and Gauss sums. These summations are exponential sums, Ramanujan sums, Dedekind sums, Kloosterman sums and their generalized sums. Besides these, the existence of some special primitive roots and the representation of some special integers are also studied. The main achievements contained in this dissertation are as follows:1. The research on exponential sums is very important in analytic number theory. In this dissertation, we study the mean value problems of two-term exponential sums and three-term exponential sums, and obtain some exact calculation formulae for their fourth and sixth power. What's more, we study the hybrid mean value related to two-term exponential sums and Dedekind sums, two-term exponential sums and polynomial character sums, and give some asymptotic formulae.2. Research on the hybrid mean value related to Dedekind sums and other summations. By using elementary and analytical methods, we study the hybrid mean value problems of Ramanujan sums and Dedekind sums, Ra-manujan sums and Hardy sums, and obtain some exact calculation formulae.3. The research on Kloosterman sums. We study the relation about classical Kloosterman sums and 2-dimensional Kloosterman sums. By using the properties of Gauss sums and Dirichlet L-functions, we study the upper bound estimate for one kind sums analogous to the high-dimensinal Kloost-erman sums, and also study the hybrid mean value of Kloosterman sums and Dedekind sums, Kloosterman sums and Hardy sums, Kloosterman sums and D. H. Lehmer problems, and get some computational formulae.4. The application of Dirichlet L-functions and Gauss sums. In this dis-sertation, we study the existence of some special primitive roots mod p, and get an asymptotic formula for the number of primitive roots satisfying some certain conditions. The estimate for character sums and the representation of some special integers are also studied.