Mean Value Formula And Integer Point Problem For Some Number Theory Functions

With development of science and technology, number theory is developing characteristically and drastically with expanding into many branches, in which many scholars from different generations have contributed a lot. The subjects that is the most concerned at present are to study the distribution of mean value and integer points of some functions in number theory as well as the exact roots for the formula including the functions of number theory. This paper is mainly focused on the arithmetic natures of some sum formulas in number theory, including Dircichlet characteristic sum function in incomplete intervals, Dirichlet L-functions, some Smaracdache functions and one type of elliptic curves. The integer roots for two equations with Smarandache function is also studied. Main results obtained in this paper are listed as follows:(1) The exact solution is developed for equations including Smarandache functions.(2) All the integer points on an elliptic are given.(3) Two asymptotic formulae of generalized quadratic Gauss sums is developed.(4) An asymptotic formula of the fourth power mean of the character sums of Dircichlet characteristic sum function in a short interval is derived by using both analytical method and elementary principles.