The Number Of Square-free Numbers In Polynomials With Smooth Variables

The theory of numbers is a branch of mathematics which deals with properties of numbers.It has a long history and has many applications in many areas.The study of asymptotic property of arithmetic function is a classical topic in number theory,and it remains a hot topic in number theory.On the other hand,representation of a sufficient large number by some type of numbers is the famous problem in number theory,for example,the Waring’s problem and The Goldbach problem.This thesis will mainly relay on square-free numbers and smooth numbers.Let n be a positive integer,if there is no prime p such that p~2|n,then n is called a square-free number.The density of square-free numbers in a subset of integers is a classical problem in analytic number theory,for example,the density of square-free number in integers,the density of square-free numbers of one variable(multi-variables)polynomials,the density of square-free numbers of one variable(multi-variables)polynomials with prime variables,the density of square-free numbers of all kinds of arithmetic functions.On the other hand,an integer n is called y-smooth,if its largest prime factor(denoted P~+(n))satisfies P~+(n)≤y,it has many similar properties with prime numbers and it plays very important roles in studying number theoretical problems,and has solved and improved many classical problems in number theory by using smooth numbers.So the study of smooth numbers is a hot topic in the area of number theory.This thesis will mainly concern on the density of squarefree numbers of polynomials with smooth variables.The main result of this thesis solves the density of polynomials of degree 1 and 2 with smooth variables,and also obtains the asymptotic formula for the representation of a large number by a square-free number and a smooth number,lastly obtains the density of square-free numbers of pairs of smooth numbers.