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

TL;DR

This paper investigates the eigenfunctions of a specific integral kernel linked to the Riemann zeta-function and Mertens function, using classical analysis and kernel theory to explore its properties.

Contribution

It initiates a general study of the eigenfunctions of the kernel involving the floor function and its connections to number theory.

Findings

Connections with the Riemann zeta-function and Mertens function

Application of classical real analysis techniques

Foundation for further spectral analysis of the kernel

Abstract

The integral kernel $K(x,y) := \frac{1}{2} + \lfloor \frac{1}{xy} \rfloor - \frac{1}{xy}$ ($0<x,y\leq 1$) 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.