Oddness of the Dedekind sum: s(k-1,k)=-s(1,k)

For a natural number $k$, write $\mathrm{dedekindSaw}(x)=0$ if the fractional part of the rational $x$ vanishes and $\mathrm{dedekindSaw}(x)=\{x\}-\tfrac12$ otherwise, and for $h\in\mathbb{Z}$ set

the sum being taken over $r$ in the range $0,\dots,k-1$ and computed in $\mathbb{Q}$. The theorem asserts, for every natural number $k$ with no further hypotheses, the identity

where the first argument is the integer $(k:\mathbb{Z})-1$, so that for $k=0$ it is $-1$ and both sides are the empty sum $0$. Thus it is the special case at $h=1$ of the combination of periodicity of $s(h,k)$ in $h$ modulo $k$ and oddness $s(-h,k)=-s(h,k)$, stated here directly for the pair $(k-1,k)$ rather than through those two general properties.

This is the classical evaluation $s(k-1,k)=-s(1,k)$ for Dedekind sums, reflecting their periodicity in the first argument and their oddness under $h\mapsto -h$. It is used in the computations of the Rademacher function $\Phi$ at particular levels, where the witnesses for residues $49$, $61$ and $109$ modulo $120$ invoke it.