I'm studying the Airy functions, and I got particularly interested in the behavior of the real zeros. I was wondering if there's some formula to express the n-th zero of the Airy functions, since they seem to have such a regular behavior, where for n-th zero I mean counting zeros in $\mathrm{Ai}(-x)$ and $\mathrm{Bi}(-x)$ from $x=0$.
Let $\zeta_n$ be the n-th zero of, for example, $\mathrm{Ai}(-x)$, I also suspect that $\zeta_n\sim \alpha n$ where $\alpha$ is some positive real constant, but reading this answer I don't think it's true at all anymore.
I thought I could use Lagrange's inversion theorem to find some formula here, given that
$$ \mathrm{Ai}(x)=\frac{1}{\pi}\int_0^\infty\cos\left(\frac{t^3}{3}+xt\right)\mathrm{d}t $$
and expanding $\cos\left(\frac{t^3}{3}+xt\right)$ using Taylor series, but I tried and it seems too difficult.
Do you have maybe even some reference to look at?
Thank you.