Study On The Properties Of Some Arithmetic Functions In Number Theory

The research on properties of famous arithmetical functions plays an im-portant role in the analytic number theory. However, because of the difficulties of the problems, the results obtained are limited. Therefore, lucubrating the problems and their generalizations is significant.In this dissertation, the properties of character sums over short intervals, a problem analogue to D.H.Lehmer's problem, a mean value of generalized Dedekind sum and Ramanujan sum and an algorithm for computing the sums of powers of integers are studied by using analytic and elementary methods in number theory. Several asymptotic formulae and identities are given. The main achievements contained in this dissertation can be described as follows:1. Research on character sums over short intervals. Firstly, a fourth power mean of Character sums over general incomplete intervals and a hybird mean value weighted by general Kloosterman sum are researched. Secondly, a hybird mean value of Character sums over quarter intervals weighted by trigonometric is studied. At last, a mean value involving Character sums, L'/L(1,χ) and Gauss sum is considered. Several asymptotic formulae are given.2. Research on a problem analogue to D.H.Lehmer's problem. Firstly, a problem analogues to D.H.Lehmer's is introduced. Then a hybird mean value between the error part of the problem and Hurwitz zeta-function is discussed and a connection between the two functions is established. A hybird mean value formula for the two varibles is given.3. Research on the generalized Dedekind sum and Ramanujan sum. Five identities involving Dedekind sum and Ramanujan sum are obtained by using Dirichlet L-function mean value identities and the properties of Character sums. The results generalize the previous conclusions.4. Research on the sums of powers of integers Sk(n). A simple technique is used to prove that S2k-1(n) is a product of n(n+1) and a rational poly-nomial of n(n+1) with degree k, and S2k(n) is a product of n(n+1)(2n+1) and a polynomial of n(n+1) with degree k. The recurrence formulae for computing the coefficients of such rational polynomials are also derived.