Additive Problems For Almost Equal Square-free Numbers

Let rv(N)denote the number of representations of the natural number N as the sum of v square-free numbers,that is the number of solutions of the equation In 1930's Evelyn and Linfoot[1]studied this problem and established asymp-totic formula for rv(N).Later Misky[4],Briidern and Perelli[5]also studied this problem,and improved the results in[1].In this paper,we shall consider the above problem for almost equal vari-ables.i.e.solvability of the equation where N??U?N,0<?<1.Let rv(n,U)denote the number of solutions of the above equation.When v?3,forU?N1/2+? and N-(v-?)U?n?N+(v-?)U,we prove that rv(n,U)=(6/?2)v(?)v(n)(?)v(n)+O(Uv-1-v-2/v-1???),where(?)v(n)is defined by(1)which satisfies(?)v(n)>>1,and(?)v(n)is defined by(2)which satisfies(?)v(n)(?)Uv-1.Specially,when U = N,we can obtain a similar result as in Briidern and Perelli[5].In this paper,we will prove above asymptotic formula by circle method.We will use the idea in Briidern and Perelli[5]and apply estimates of expo-nential sum over square-free numbers in short intervals.This thesis is divided into four chapters.In the first chapterwe introduce the background and history of the problem,and give the asymptotic formula for rv(n,N).In the second chapter,we introduce the circle method and give the proof of the main result.In the third chapter,we estimate the integrals over the major arcs and the minor arcsIn the last chapter,we estimate the exponential sum over square-free numbers in short interval,and give proofs of other related lemmas.