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

Abstract

We show that the kernel $K(x,y)=\frac{1}{2}+\lfloor \frac{1}{xy}\rfloor -\frac{1}{xy}$ ($0<x,y\le 1$) has infinitely many positive eigenvalues and infinitely many negative eigenvalues. Our interest in this kernel is motivated by the appearance of the quadratic form ${\sum }_{m,n\le N}K(\frac{m}{N},\frac{n}{N})\mu (m)\mu (n)$ in an identity involving the Mertens function.

8 Pages, Plain Tex, submitted to the Journal de Théorie des Nombres de Bordeaux. Revised to 9 Pages (Plain Tex), following the anonymous referee's comments. Main changes: paragraphs concerning eigenfunctions and an identity that was obtained by F. Mertens in 1897 have been added to Section 1, and details of the relevant paper of Mertens have been included in the references