On The Average Number Of Subgroups Of The Group Z_m × Z_n

In this dissertation,we mainly consider several problems about the mean value estimation of the number of subgroups of group Z_m × Z_n,such as the average number of subgroups of group Z_m × Z_n with mn ≤x,the average number of subgroups of group Z_m × Z_n with m²+n² ≤x,the mean value of the square of the number of subgroups and the average number of subgroups of the group Zm× Zn with k-th power sequence,and the average number of subgroups of the group Zm₁× Zm₂ with m₁m₂≤x such that m₁m₂ is a k-th power.There exist total five chapters in this dissertation.We list the contents of all the chapters as follows:In Chapter one,we introduce the history and the development of the problems.Also,current results are listed.Meanwhile,we give our improvement of the results.In Chapter two,We use the method of dealing with the multidimensional weighted divisor problem to deal with the mean value of the number of subgroups of group Z_m × Z_n with mn ≤x.First,we use the traditional Perron formula method to obtain a weaker result of this problem,and then we improve previous result by more delicate estimates of exponential sums.In addition,the upper bound of the mean-square estimate for the error term of the asymptotic formula obtained from the above problem is also given in this chapter.In Chapter three,we use elementary method and exponential sum estimation method to deal with the mean value of the number of subgroups of group Zm ×Zn with m²+n² ≤x.The large sieve inequality is introduced to deal with the main part to give the asymptotic formula of this problem.This problem can be considered as a generalization of both the Dirichlet divisor problem and the circle problem.In Chapter four,we consider the mean value of the square of the number of subgroups and the average number of subgroups of the group Z_m × Z_n with k-th power sequence,and obtain asymptotic formulas for the above two problems by using a multidimensional Perron formula and the complex integration method.In Chapter five,we consider the average number of subgroups of the group Zm₁×Zm₂ with m₁m₂≤x such that m₁m₂ is a k-th power number,and mainly study two special cases:m₁m₂ is a square number and a cubic number.We get the asymptotic formula for both cases by the multidimensional Perron formula method and the complex integration method.