On the density of Sylow numbers

Abstract

Let $p$ be a prime number. We say that a positive integer $n$ is a Sylow $p$-number if there exists a finite group having exactly $n$ Sylow $p$-subgroups. When $p=2$, every odd integer is a Sylow $2$-number. In contrast, when $p$ is odd, there exist two positive constants $c_p$ and $c_p^\prime$ such that, denoting by $\beta(p,x)$ the number of Sylow $p$-numbers less than or equal to $x$, \[c_p\,x(\log x)^{\frac{1}{p-1}-1} \leq \beta(p,x)\leq c_p^\prime\,x(\log x)^{\frac{1}{p-1}-1}. \] Moreover if $\beta_s(p,x)$ is the number of positive integers $n\le x$ such that $n$ is the Sylow $p$-number of some finite solvable group then $$\beta_s(p,x)\sim c_p\,x(\log x)^{\,\frac{1}{p-1}-1} \qquad\text{as } x\to\infty.$$ In particular, when $p$ is odd, the natural density of Sylow $p$-numbers is $0$.