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We study the eigenvalues $\lambda_1,\lambda_2,\lambda_3,\ldots$ (ordered bymodulus) of the integral kernel $K(x,y) := \frac{1}{2} + \lfloor \frac{1}{xy}\rfloor - \frac{1}{x y}$ ($0<x,y\leq 1$). This kernel is of interest inconnection with an identity of F. Mertens involving the M\"obius function. Weestablish that $\sum_{m=1}^{\infty} |\lambda_m|^{-1} = \infty$, and prove that$|\lambda_m| > m\log^{-3/2} m$ for all but finitely many positive integers $m$.The first of these results is an application of the theory of Hankel operators;the proof of the second result utilises a family of degenerate kernels$k_3,k_4,k_5,\ldots$ that are step-function approximations to $K$. Throughseparate computational work on eigenvalues of $k_N$ ($N=2^{21}$) we obtainnumerical bounds, both upper and lower, for specific eigenvalues of $K$.Further computational work, on eigenvalues of $k_N$ ($N\in\{2^{10},2^{11},\ldots ,2^{21}\}$), leads us to formulate a quite preciseconjecture concerning where on the real line the eigenvalues$\lambda_1,\lambda_2,\ldots ,\lambda_{767}$ are located: we discuss how thisconjecture could (if it is correct) be viewed as supportive of certaininteresting general conjectures concerning the eigenvalues of $K$.

On eigenvalues of the kernel $\frac{1}{2} + \lfloor \frac{1}{xy}\rfloor - \frac{1}{xy}$, II