Pair correlations of logarithms of integers

TL;DR

This paper investigates the pair correlations of logarithms of positive integers under various scalings and weights, revealing phenomena like level repulsion, mass loss, and Poissonian behavior, with applications to geodesic lengths in modular curves.

Contribution

It establishes the existence of pair correlation functions for logarithms of integers and analyzes their behavior under different scalings and weights, including applications to modular curves.

Findings

Pair correlations exhibit level repulsion at linear scaling.

Mass loss occurs at superlinear scalings.

Poissonian behavior is observed at sublinear scalings.

Abstract

We study the correlations of pairs of logarithms of positive integers at various scalings, either with trivial weigths or with weights given by the Euler function, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to itself in the modular curve $\operatorname{PSL}_2(\mathbb Z)\backslash\mathbb H^2_{\mathbb R}$.