正整数积性子半群中的计数问题

Counting problem in multiplicative subsemigroups of positive integers

设S和S'为正整数集N满足特定条件的乘子半群的最小生成元系,记〈A〉为由A生成的乘子半群,以及N_A(x):=∑n∈〈A〉:n≤x1.使用初等的求和换序方法得到了一个建立N_S(x)和N_(S∪S')(x)联系的计算公式.利用该公式以及多变量的数学归纳法推出了由有限递增素数列{p_i}生成的子半群中元素个数的渐近估计式.

Let S and S＇ be minimal systems of generators of specific subsemigroups of positive integers N. If A N, then 〈A〉 is the subsemigroup generated by A. Let NA（x）：=∑n∈〈A〉：n≤x1. A formula that establishes a connection between Ns（x） and Nsus,（x） is ob-tained via elementary methods of changing summation order. With this formula and induction on several variables, an asymptotic estimation of the number of element in a subsemigroup generated by a finite set of primes is obtained.