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

Abstract

The integral kernel $K(x,y):=\frac{1}{2}+\lfloor \frac{1}{xy}\rfloor -\frac{1}{xy}$ ($0<x,y\le 1$) has connections with the Riemann zeta-function and a (recently observed) connection with the Mertens function. In this paper we begin a general study of the eigenfunctions of $K$. Our proofs utilise some classical real analysis (including Lebesgue's theory of integration) and elements of the established theory of square integrable symmetric integral kernels.

58 pages, LaTeX. Submitted to Functiones et Approximatio Commentarii Mathematici