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

Abstract

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.