The group ring of Q/Z and an application of a divisor problem

TL;DR

This paper establishes elementary identities in the group ring of Q/Z and applies them, along with analytic number theory, to estimate the size of specific sets related to Farey fractions and divisor problems.

Contribution

It introduces new identities in the group ring of Q/Z and demonstrates their application to problems involving Farey fractions and divisor estimates.

Findings

Identities in the group ring of Q/Z are proven.

Applications to estimating sets related to Farey fractions.

Connections to divisor problems in short intervals.

Abstract

First we prove some elementary but useful identities in the group ring of Q/Z. Our identities have potential applications to several unsolved problems which involve sums of Farey fractions. In this paper we use these identities, together with some analytic number theory and results about divisors in short intervals, to estimate the cardinality of a class of sets of fundamental interest.