The $n\times n$ Hilbert matrix $H$ is defined as

$H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$

What is known about the singular values $\sigma_1\geq\ldots\geq \sigma_n$ of $H$?

For example, it is known that the matrix is very ill-conditioned, i.e., $\sigma_1/\sigma_n = \mathcal{O}((1+\sqrt{2})^{4n}/\sqrt{n})$ [1].

But are there estimates for $\sigma_k$ for $2\leq k\leq n-1$? Or is there a bound on $1\leq k\leq n$ such that $\sigma_k > \epsilon$ for some $\epsilon>0$? I'm interested in this problem because I would like to know the numerical rank of $H$.

[1] J. Todd, "The condition of the finite segments of the Hilbert matrix." Contributions to the solution of systems of linear equations and the determination of eigenvalues, 39 (1954), pp. 109-116.