Research On The Arithmetical Properties Of Some Summations In Number Theory

The main purpose of this dissertation is to study the arithmetical properties of some summations in number theory. These summations are Character sums over incomplete intervals, Dirichlet characters of polynomials in several variables, Exponential sums with characters, hyper-Kloosterman sums, Dedekind sums and their generalized sums. Besides these, D. H. Lehmer problems and some Smarandache type functions were also studied. The main achievements contained in this dissertation are follows:1. Studied the hig.her power mean of complete trigonometric sums and two term exponential sums with Dirichlet characters, and obtained some exact calculation formulas for their fourth power mean by using analytical and elementary methods;2. Studied the asymptotic properties of higher power mean of Dirichlet character sums over incomplete intervals. At first, I studied the odd and even primitive character sums over quarter interval, respectively, and obtained some asymptotic formulas of their 2κth power mean; Secondly, I studied the 2κth power mean of even primitive character sums over one eighth interval and got a sharper asymptotic formula for it; Thirdly, I studied the fourth power mean of nonprincipal character sums over quarter interval and obtained a sharper asymptotic formula too; Fourthly, I studied the first power mean of primitive character sums over quarter interval and got some asymptotic formulas for it; At last, applying the identity for character sums over quarter interval which I obtained, generalized and proved the famous Euler numbers conjecture;3. Studied the evaluation problem of Dirichlet characters of polynomial in several variables and obtained some identities for a kind of Dirichlet characters of polynomial. This results explained the fact that Kats's estimation is the best possible;4. Defined some special mean values for the classical Dedekind sums and its analogous sums which called Cochrane sums, and studied their asymptotic properties. Defined the high dimensional Cochrane sums and obtained its order estimation and asymptotic formula of its square mean value;5. Studied the higher power mean of hyper-Kloosterman sums and got an calculation formula for their fourth power mean under some restrictions;6. Studied a mean value of error term of D. H. Lehmer problem and obtained some interesting and strange results. Restricted D. H. Lehmer problem on a half interval. Studied the error term of the D. H. Lehmer problem over half interval and obtained a sharper asymptotic formula for its square mean value;7. Studied the value distribution of Smarandache function and the mean value of Smarandache power function, and obtained some interesting properties.