Dedekind sums in the $p$-adic number field

TL;DR

This paper investigates the density of Dedekind sums in the field of p-adic numbers, showing they are dense for primes p ≥ 5 but not for p=2 or 3.

Contribution

It provides a complete answer to Kohnen's question by characterizing the density of Dedekind sums in p-adic fields based on the prime p.

Findings

Dedekind sums are dense in rac{p}{p} for p ge 5.

Dedekind sums are not dense in rac{2}{2} and rac{3}{3}.

Dedekind sums do not approximate units in rac{rac{Z}{2}} and rac{rac{Z}{3}}.

Abstract

In a recent note W. Kohnen asks whether the values of Dedekind sums are dense in the field of $p$-adic numbers. The present paper answers this question. Dedekind sums do not approximate units of $\mathbb Z_2$ or $\mathbb Z_3$, so they are not dense in $\mathbb Q_2$ or $\mathbb Q_3$. But they are dense in $\mathbb Q_p$ if $p\ge 5$.