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

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

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

For example, it is known that the matrix is very ill-conditioned, i.e., [1]

$$\dfrac{\sigma_1}{\sigma_n} = \mathcal O \left( \frac{1+\sqrt{2})^{4n}}{\sqrt{n}} \right)$$

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 am 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.