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Abstract
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We study the statistics of pairs of logarithms of positive integers at various scalings,
either with trivial weights or with weights given by the Euler function, proving the
existence of pair correlation functions. We prove that at the linear scaling, which is
not the usual scaling by the inverse of the average gap, the pair correlations exhibit a
level repulsion similar to radial distribution functions of fluids. We prove total loss of
mass phenomena at superlinear scalings, and constant nonzero asymptotic
behavior 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 .
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Keywords
pair correlation, logarithms of integers, level repulsion,
Euler function
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Mathematical Subject Classification
Primary: 11J83, 11K38, 11N37, 53C22
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Milestones
Received: 30 August 2021
Revised: 14 June 2022
Accepted: 20 July 2022
Published: 25 November 2022
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