A Study Of Two Types Of Special Number Sets In Analytic Number Theory

The property of integers is one of the main research objects of analytic number theory.It has a very rich content and a long history.Many scholars have done indepth research on it,and got a lot of meaningful conclusions.The research on the number sets with various special properties in the integer set has become an important direction of the research on the integer set,which also produces many famous problems,including Riemann hypothesis,Goldbach conjecture,Waring problem,Twin prime conjecture and so on.Therefore,as two special number sets in integer set,cube-full numbers and D.H.Lehmer numbers are of great significance to the development of analytic number theory.Based on the research of many scholars,this paper mainly studies the distribution of cube-full numbers in arithmetic progression,and obtains an asymptotic formula by using elementary and analytic methods.Secondly,this paper also studies the problem that integer N can be expressed as the sum of three generalized Lehmer numbers:1.For cube-full numbers:Let P3 is the set of cube-full numbers,we study the sum in arithmetic sequence n ≡l（mod q）,and obtain the asymptotic formula correspondingly.2.For Lehmer numbers:We define the generalized Lehmer numbers as follows:Let q is an integer,n and c are two fixed integers with n≥ 2,q>n,（c,q）=（n,q）=1.For 1≤a≤q with（a,q）=1,we denote by b the unique integer 1 ≤b 14q satisfies b≡ cam（mod g）,and let L（q）={a∈Z+:（a,q）=1,n（?）a+b｝.Then we give the asymptotic formula for the number of representations of integer N as the sum of three generalized Lehmer numbers.