On the moduli of a Dedekind sum

TL;DR

This paper constructs a sequence of pairs (a_r, b_r) with polynomial b_r of degree 4, demonstrating the moduli of Dedekind sums can be realized with polynomial growth in the denominator.

Contribution

It provides a new explicit polynomial sequence of degree 4 for the denominators of Dedekind sums with fixed rational values.

Findings

Sequence with polynomial degree 4 for denominators

Explicit construction of Dedekind sum values

Polynomial growth in denominators

Abstract

Let $s(a,b)$ denote the classical Dedekind sum and $S(a,b)=12s(a,b)$. Let $k/q$, $q\in \Bbb N$, $k\in \Bbb Z$, $(k,q)=1$, be the value of $S(a,b)$. In a previous paper we showed that there are pairs $(a_r,b_r)$, $r\in\Bbb N$, such that $S(a_r,b_r)=k/q$ for all $r\in \Bbb N$, the $b_r$'s growing in $r$ exponentially. Here we exhibit such a sequence with $b_r$ a polynomial of degree $4$ in $r$.