For a research problem I am working on in PDE, I need to obtain asymptotics for the counting function of $$\{0<\alpha <\lambda: \exists n\in \mathbb{N} \textrm{ such that }J\_n(\alpha)=0 \textrm{ and }\alpha<Cn \}$$ where $C>1$ is fixed.


Has anyone come across any techniques or prior results well suited to this question?



The asymptotic distribution of a subset of Bessel function zeroes
Let $\alpha=j\_{n,m}$ be the positive zeros of $J\_n(\alpha)$, numbered so that $j\_{n,m+1} > j\_{n,m}$. Let $m\_{n,\alpha}$ denote the largest $m$ such that $j\_{n,m} < \alpha$. It seems to me that you want to asymptotically estimate the sum
$$N\_\lambda = \sum\_{n<\lambda/C} m\_{n,Cn}.$$
Since the numbers $m\_{n,Cn}$ would be increasing, the asymptotics should be dominated by the largest term $m\_{\lambda/C,\lambda}$. To get that, you need to solve the equation $j\_{\lambda/C,m} = \lambda$ for $m$ as $\lambda \to \infty$. There is an asymptotic expansion for $j\_{n,m}$ for large $n$ that is uniform in $m$ ([DLMF, §10.21.41](http://dlmf.nist.gov/10.21.E41)), which is what I think should be used here. If I have decoded this uniform approximation correctly, then
$$ m\_{\lambda/C,\lambda} \sim K \lambda, \quad \text{with} \quad
 K = \frac{\sqrt{C^2-1}-\sec^{-1}C}{\pi C} . $$
So your estimate should be $N\_\lambda \sim K \lambda^2/2$.


