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

We study the eigenvalues ${\lambda }_1,{\lambda }_2,{\lambda }_3,\dots$ (ordered by modulus) of the integral kernel $K(x,y) := \frac{1}{2} + \lfloor \frac{1}{x y}\rfloor - \frac{1}{x y}$($0<x,y\leq 1$). This kernel is of interest in connection with an identity of F. Mertens involving the M"obius function. We establish that ${\sum }_{m=1}^{\mathrm{\infty }}\mathrm{\mid }{\lambda }_m{\mathrm{\mid }}^{-1}=\mathrm{\infty }$, and prove that $\mathrm{\mid }{\lambda }_m\mathrm{\mid }>m{\log }^{-3\mathrm{/}2}m$ 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 $k_3,k_4,k_5,\dots$ that are step-function approximations to $K$. Through separate computational work on eigenvalues of $k_N$ ($N=2^{21}$) we obtain numerical 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 precise conjecture concerning where on the real line the eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{767}$ are located: we discuss how this conjecture could (if it is correct) be viewed as supportive of certain interesting general conjectures concerning the eigenvalues of $K$.