  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 $\operatorname{PSL}_2(\mathbb
Z)\backslash\mathbb H^2_{\mathbb R}$.
