On eigenvalues of the kernel $\frac{1}{2}+\lfloor \frac{1}{xy}\rfloor -\frac{1}{xy}$ ($0<x,y\le 1$)

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 and infinitely many negative eigenvalues. Our interest in this kernel is motivated by 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.