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

Abstract

We study the eigenvalues $\lambda_1,\lambda_2,\lambda_3,\ldots$ (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 $\sum_{m=1}^{\infty} |\lambda_m|^{-1} = \infty$, and prove that $|\lambda_m| > m\log^{-3/2} m$ 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 $k_3,k_4,k_5,\ldots$ 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 $\lambda_1,\lambda_2,\ldots ,\lambda_{767}$ 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 $K$.