Invariance of s(h,k) under inversion modulo k

Let $h$, $h'$ and $k$ be natural numbers such that $h h' \equiv 1 \pmod{k}$. Here, for an integer $a$ and a natural number $m$, the Dedekind sum is defined by

a rational number, where the sawtooth function is given on a rational $x$ by $((x)) = 0$ if the fractional part of $x$ vanishes and $((x)) = \{x\} - \tfrac12$ otherwise, with $\{x\}$ the fractional part; for $m = 0$ the sum is empty, hence $0$. The conclusion is the equality of rationals $s(h',k) = s(h,k)$, the arguments $h$ and $h'$ being taken as integers via the canonical map from $\mathbb{N}$. Thus the Dedekind sum is unchanged when the first argument is replaced by an inverse of it modulo the second; no positivity, coprimality or size hypotheses beyond the stated congruence are imposed.

This is the classical invariance $s(h',k) = s(h,k)$ for $h h' \equiv 1 \pmod k$, one of the elementary arithmetic properties of Dedekind sums alongside periodicity in $h$ and the reciprocity law. It is used in the evaluation of a Rademacher $\Phi$-type level witness, in rademacher_phi_level_witness_mod_oneTwenty_eq_sixtyOne.