# The Petersson-Knopp identity and Farey neighbours

## TL;DR

This paper investigates the behavior of Dedekind sums near Farey points, revealing how the Petersson-Knopp identity relates sums close to expected values and their frequency of occurrence.

## Contribution

It provides a new interpretation of the Petersson-Knopp identity in the context of Dedekind sums near Farey points, linking sums to expected values and their frequencies.

## Key findings

- Dedekind sums near Farey points are closely related to expected values.
- The Petersson-Knopp identity explains the frequency of these sums occurring near expected values.
- A specific interpretation of the Petersson-Knopp identity in this setting is established.

## Abstract

We study Dedekind sums $S(a,b)$ near Farey points of the interval $[0,b]$. Each of these Dedekind sums is connected with a set of other Dedekind sums by the Petersson-Knopp identity. In the case considered here, this identity has a very specific interpretation, inasmuch as each Dedekind occurring in this identity is close to a certain expected value. Conversely, each of these expected values occurs with a certain frequency, a frequency that is consistent with the Petersson-Knopp identity.