Numbers which are orders only of cyclic groups

Abstract:

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 $\gcd (n,\phi (n))=1$. With $C(x)$ denoting the count of cyclic $n\le x$, Erdős proved that \[ C(x) \sim e^{-\gamma } x/\!\log \log \log {x}, \quad \text {as $x\to \infty $}. \] We show that $C(x)$ has an asymptotic series expansion, in the sense of Poincaré, in descending powers of $\log \log \log {x}$, namely \[ \frac {e^{-\gamma } x}{\log \log \log {x}} \left (1\!-\!\frac {\gamma }{\log \log \log {x}} \!+\! \frac {\gamma ^2 + \frac {1}{12}\pi ^2}{(\log \log \log {x})^2} \!-\! \frac {\gamma ^3 +\frac {1}{4} \gamma \pi ^2 \!+\! \frac {2}{3}\zeta (3)}{(\log \log \log {x})^3} + \dots \right ). \]