The Petersson-Knopp identity and Farey neighbours

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.