Research On The Class Dedekind And Its Mean

There are some famous sums, such as Dedekind sums, Hardy sums, Klooster-man sums, character sums, whose mean value and upper bound of a single sum play an important role in the study of analytic number theory. Many scholars at home and abroad have conducted the extensive and thorough research on these questions, and have obtained some abundant results. The mean value and the upper bound of a single sum have a very close relation to many well known open problems or im-portant conjectures. Therefore, any relevant progress is promoting the development of analytic number theory.Based on our interests in the above problems, this thesis is aimed at studying the hybrid mean value problem involving Hardy sums and Kloosterman sums by using the elementary and analytic methods, and getting some interesting hybrid mean value formulae. Moreover, we define a new incomplete Cochrane sums, and try to give a upper bound estimate. Specially, the main achievements contained in this thesis are as follows:1. The hybrid mean value involving Hardy sums S4(mn,p) and Kloosterman sums and the hybrid mean value involving Hardy sums S-,(mn,p) and Kloosterman sums are studied. By using the properties of Gauss sums and the mean value theorem of the Dirichlet L-function, some interesting results are obtained.2. A new sum (incomplete Cochrane sums) is defined. By using the properties of Kloosterman sums and Dirichlet character, the upper bound estimate of incomplete Cochrane sums is shown.