Additive Problems On Smooth Number

Let y ≥ 2 be a real number.n is called a y-smooth number,if the largest prime factor of n is less than or equal to y.The smooth number is an important branch of number theory,and the additive problem of smooth numbers is an important research object of additive number theory.In this thesis,we mainly study the additive problem of smooth numbers.Firstly,in classical additive number theory,Walfisz studied the asymptotic formula of the number of ways of writing n as a sum of a prime and square-free number.Romanoff studied the lower bound of the number of integers that can be expressed as a sum of a prime and a power of 2.Based on these theorems,we studies the similar Romanoff theorem and Walfisz theorem in smooth numbers.More precisely,we mainly studies two problems.One is that the asymptotic formula about the number of ways of writing n as a sum of smooth number and square-free number.It can be experssed as the product of the number of smooth numbers and the density without square factors up to a constant.The second result is the lower bound of the number of integers that can be expressed as a sum of a smooth number and a powers of a,which is Cρ(u)2/5++o(ε),where C is a constant,ρ(u)≤ 1 is the Dickman function.In the proof of the first result,we employ the distribution results of smooth numbers in arithmetic progressions and the properties of the Mobius function.To prove second result we apply the upper bound of density of the polynomial sequence{Q(n)} determine the in y-smooth numbers,the Cauchy inequality.In addition,under Martin’s conjecture,this lower bound can be improved better.The additive problem of smooth numbers is an extension of the Goldbach conjecture and the Waring problem.The results of this thesis will develop the additive number theory.It also plays an important role in the study of geometry,probability and other fields.