论文标题
用规定的子组计数乘法组
Counting multiplicative groups with prescribed subgroups
论文作者
论文摘要
我们研究了两个计数问题，这些问题在表面上似乎是非常群体理论的，但是在仔细检查时，发现与整数有关其主要因素的限制。 首先，考虑到一个奇怪的Prime $ Q $和有限的Abelian $ Q $ -Group $ H $，我们考虑了一组整数$ n \ le X $，使得Sylow $ q $ -subgroup的乘法组的$（\ Mathbb z/n \ mathbb Z）我们表明，这组整数的计数函数是渐近常数$ k x（\ log \ log x）^\ ell/（\ log x）^\ ell/（\ log x）^{1/（q-1）} $，对于显式常数$ k $，$ k $和$ \ ell $，根据$ q $和$ h $。 其次，我们考虑一组整数$ n \ le x $，以便乘法组$（\ mathbb z/n \ mathbb z）^\ times $是“最大非循环的”，也就是说，其所有Prime-Power子组都是基本组。我们表明，对于显式常数$ a $，这组整数的计数函数是渐近的，$ a x/（\ log x）^{1-ξ} $，其中$ξ$是Artin的常数。 事实证明，这两个群体理论问题都可以简化为对整数限制其主要因素的问题，从而可以通过分析数理论的经典技术来解决它们。
We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime $q$ and a finite abelian $q$-group $H$, we consider the set of integers $n\le x$ such that the Sylow $q$-subgroup of the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ is isomorphic to $H$. We show that the counting function of this set of integers is asymptotic to $K x(\log\log x)^\ell/(\log x)^{1/(q-1)}$ for explicit constants $K$ and $\ell$ depending on $q$ and $H$. Second, we consider the set of integers $n\le x$ such that the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ is "maximally non-cyclic", that is, such that all of its prime-power subgroups are elementary groups. We show that the counting function of this set of integers is asymptotic to $A x/(\log x)^{1-ξ}$ for an explicit constant $A$, where $ξ$ is Artin's constant. As it turns out, both of these group-theoretic problems can be reduced to problems of counting integers with restrictions on their prime factors, allowing them to be addressed by classical techniques of analytic number theory.
