$f(n)$ is the number of groups of order $n$ up to isomorphism.
I am reading this article. There is a section called "Great gnus". In this section, there is an asymptotic estimate of $f(n)$. $$p^{\frac{2}{27}n^{3}+O\left(n^{8/3}\right)}$$
What it means?
In my head it means that for every positive constant $k$
$$f(p^{n}) \leq p^{\frac{2}{27}n^{3}+k\cdot n^{8/3}}$$
for sufficient large $n$. Am I correct?
Standard sense: there exists $g(n)$ such that asymptotic estimate of $f(n)$ is $$p^{\frac{2}{27}n^{3} + g(n)}$$ and $g(n) = O\left(n^{8/3}\right)$.
Moreover, $g(n) = O\left(n^{8/3}\right)$ iff $|g(n)| \le k n^{8/3}$ for some constant $k$.
Here is a possible interpretation:
$$ f(n) = a^{g(n)+O(h(n))} \; \Leftrightarrow \; \log_a f(n) - g(n) = O(h(n)) $$ as $ n \to \infty$. This is equivalent to $$ \exists \, C\, \forall \, n \, \frac{f(n)}{a^{g(n)}} \le C^{h(n)} \, . $$
