Numbers which are orders only of cyclic groups

TL;DR

This paper studies the distribution of cyclic numbers, showing that the count of such numbers up to x has an asymptotic series expansion involving iterated logarithms, refining previous asymptotic results.

Contribution

It derives an asymptotic series expansion for the count of cyclic numbers, providing a detailed approximation beyond the leading term.

Findings

Asymptotic series expansion for $C(x)$ in powers of $1/\log\log\log{x}$

Refinement of Erd ext{"o}s' asymptotic estimate for cyclic numbers

Explicit coefficients involving Euler's constant, $\pi$, and $\zeta(3)$

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 $n\le x$, Erd\H{o}s 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 Poincar\'e, in descending powers of $\log\log\log{x}$, namely $$\frac{e^{-\gamma} x}{\log\log\log{x}} \left(1-\frac{\gamma}{\log\log\log{x}} + \frac{\gamma^2 + \frac{1}{12}\pi^2}{(\log\log\log{x})^2} - \frac{\gamma^3 +\frac{1}{4} \gamma \pi^2 + \frac{2}{3}\zeta(3)}{(\log\log\log{x})^3} + \dots \right). $$