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

10 pages; some typos corrected