论文标题
数字仅是循环组的订单
Numbers which are orders only of cyclic groups
论文作者
论文摘要
如果每组订单$ n $都是循环的，我们将$ n $称为循环号码。它隐含在迪克森（Dickson）的作品中，并在Szele的工作中明确说明，$ n $是$ \ gcd（n，ϕ（n））= 1 $时的循环。用$ c（x）$表示周期性$ n \ le x $的计数，erdős证明了$ c（x）\ sim e^{ - γ} X/\ log \ log \ log \ log \ log \ log {x}，\ quad \ quad \ text {as $ x \ to \ x \ to \ for \ infty $}。 poincaré，在$ \ log \ log \ log \ log {x} $的降序中，即$$ \ frac {e^{ - γ} x} {\ log \ log \ log \ log \ log {x}}}} \ left（1- \fracγ{\ log \ log \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ frac {x}} \ frac {1} {12}π^2} {（\ log \ log \ log \ log {x}）^2} - \ frac {γ^3 + \ frac {1} {4}γπ^2 + \ frac + \ frac {2} {2} {2} {3} {3} {3） \ dots \ right）。 $$
We call $n$ a cyclic number if every group of order $n$ is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that $n$ is cyclic precisely when $\gcd(n,ϕ(n))=1$. With $C(x)$ denoting the count of cyclic $n\le x$, Erdős proved that $$C(x) \sim e^{-γ} x/\log\log\log{x}, \quad\text{as $x\to\infty$}.$$ We show that $C(x)$ has an asymptotic series expansion, in the sense of Poincaré, in descending powers of $\log\log\log{x}$, namely $$\frac{e^{-γ} x}{\log\log\log{x}} \left(1-\fracγ{\log\log\log{x}} + \frac{γ^2 + \frac{1}{12}π^2}{(\log\log\log{x})^2} - \frac{γ^3 +\frac{1}{4} γπ^2 + \frac{2}{3}ζ(3)}{(\log\log\log{x})^3} + \dots \right). $$
