A note on the pair correlation of Farey fractions

TL;DR

This paper investigates the pair correlation statistics of Farey fractions with specific denominator conditions, providing explicit formulas for their distribution patterns.

Contribution

It explicitly computes the pair correlation functions for Farey fractions under certain coprimality and congruence conditions, extending previous understanding.

Findings

Pair correlation functions exist for Farey fractions with given denominator restrictions.

Explicit formulas for these correlation functions are derived.

Results enhance understanding of Farey sequence distribution patterns.

Abstract

The pair correlations of Farey fractions with denominators $q$ satisfying $(q,m)=1$, respectively $q\equiv b \pmod{m}$ with $(b,m)=1$, are shown to exist and are explicitly computed.