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On the statistics of pairs of logarithms of integers

Jouni Parkkonen and Frédéric Paulin

Appendix: Étienne Fouvry

Vol. 11 (2022), No. 4, 335–372
DOI: 10.2140/moscow.2022.11.335
Abstract

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  PSL 2()2.

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Keywords
pair correlation, logarithms of integers, level repulsion, Euler function
Mathematical Subject Classification
Primary: 11J83, 11K38, 11N37, 53C22
Milestones
Received: 30 August 2021
Revised: 14 June 2022
Accepted: 20 July 2022
Published: 25 November 2022
Authors
Jouni Parkkonen
Department of Mathematics and Statistics
University of Jyväskylä
Jyväskylä
Finland
Frédéric Paulin
Laboratoire de mathématique d’Orsay
UMR 8628 CNRS
Université Paris-Saclay
Orsay
France
Étienne Fouvry
Laboratoire de mathématique d’Orsay
UMR 8628 CNRS
Université Paris-Saclay
Orsay
France