Research On The Arithmetical Function ?(n) And Certain Diophantine Equations

In this thesis,we mainly study the probability distribution of the number theory function ?(n)and the number of solutions to certain infinite Diophantine equations,then we give their generating functions and asymptotic formulas.The specific results are as follows:In the first part,let ?(n)be the number of distinct prime factors of the natural number n,we mainly discuss the probability distribution of the number theory function ?(n).In the 1930s,Erd(?)s and Kac studied the distribution of ?(n)in the interval[2,x],where x is a sufficiently large natural number.And the famous Erdos-Kac theorem is obtained,which is considered as the second birth certificate of probabilistic number theory.In this paper,using the complex analysis method,we extend the Erdos-Kac theorem to the higher order and more general form,and improve Granville and Soundarajan's work in this field.In the second part,for any given positive integers n,r,we study the following infinite Diophantine equations:n=1r·|k1|v+2r·|k2|v+3r·|k3|v+…,where k=(k1,k2,k3,…)?Z?,v?{1,2}.We use Sr,v(n)to denote the number of solutions of the above infinite Diophantine equation.Based on the basic asymptotic analysis and some basic techniques from analytic number theory,we obtain the generating functions and asymptotic formulas of Sr,v(n)by using the integer partitions and the circle method.