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We show that the kernel $K(x,y)={1\over 2}+\lfloor {1\over xy}\rfloor-{1\over xy}$ ($0<x,y\leq 1$) has infinitely many positive eigenvalues andinfinitely many negative eigenvalues. Our interest in this kernel is motivatedby the appearance of the quadratic form $\sum_{m,n\leq N} K\bigl( {m\over N} ,{n\over N}\bigr) \mu(m)\mu(n)$ in an identity involving the Mertens function.

On eigenvalues of the kernel ${1\over 2}+\lfloor {1\over xy}\rfloor - {1\over xy}$ ($0