The exercise in question (which can be found in page 71) asks to derive the reciprocity law for the Dedekind sums from the functional equation for $\eta(\tau)$, where $\eta$ is Dedekind's eta function. I tried this exercise without success. I would appreciate if anyone could write a solution or provide some hints.
Dedekind's functional equation (Apostol Theorem 3.4) says that, for all $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in \text{SL}_2(\mathbb Z)$ with $c>0$ and $\tau\in \mathbb H$, we have $$ \eta\left(\frac{a\tau+b}{c\tau+d}\right)=\varepsilon(a,b,c,d)\sqrt{-i(c\tau+d)}\eta(\tau) $$ where $$ \varepsilon(a,b,c,d)=\exp\left\{ \pi i \left(\frac{a+d}{12c}+s(-d, c)\right)\right\} $$ and $$ s(h, k)=\sum_{r=1}^{k-1}\frac{r}{k}\left(\frac{hr}{k}-\bigg\lfloor\frac{hr}{k}\bigg\rfloor-\frac 12\right). $$ A stronger equation holds (this is Equation (12) referenced by Apostol): \begin{gather} \tag{12} \log\eta\left(\frac{a\tau+b}{c\tau +d}\right)=\log\eta(\tau)+\pi i\left(\frac{a+d}{12 c}+s(-d, c)\right)+\frac 12\log(-i(c\tau +d)). \end{gather}
The $s(h,k)$ are the Dedekind sums.
The reciprocity law for the Dedekind sums (Apostol Theorem 3.7) says $$ 12hks(h, k)+12khs(k,h)=h^2+k^2-3hk+1 $$ for $h, k$ coprime positive integers.
My attempt: In a nutshell, what I tried to do is to express $s(h, k)$ and $s(k, h)$ via $\eta$ using the functional equation. This leaves several "free" variables that are independent of $h$ and $k$, and then I was hoping that by some wise substitution the reciprocity law would be immediate. More specifically, we have $$ s(-d, c)=\frac{1}{\pi i}\log \eta\left(\frac{a\tau+b}{c\tau+ d}\right)-\frac{1}{\pi i}\log\eta(\tau)-\frac{a+d}{12 c}-\frac{1}{2\pi i}\log(-i(c\tau +d)). $$ If now $x,y,x', y'\in\mathbb Z$ are such that $$ \begin{pmatrix} x&y\\ k&-h \end{pmatrix} , \begin{pmatrix} x'&y'\\ h&-k \end{pmatrix} \in \text{SL}_2(\mathbb Z) $$ then, for $\tau, \tau'\in \mathbb H$, \begin{gather*} s(h,k)+s(k,h)=\frac{1}{\pi i}\log\eta\left(\frac{x\tau+y}{k\tau-h}\right)-\frac{1}{\pi i}\log \eta(\tau)-\frac{x-h}{12k}-\frac{1}{2\pi i}\log(-i(k\tau-h))+\\ +\frac{1}{\pi i}\log\eta\left(\frac{x'\tau'+y'}{h\tau'-k}\right)-\frac{1}{\pi i}\log \eta(\tau')-\frac{x'-k}{12h}-\frac{1}{2\pi i}\log(-i(h\tau'-k)), \end{gather*} and this should be equal to $$ \frac{h}{12k}+\frac{k}{12h}-\frac 14+\frac{1}{12hk}. $$ We can choose $x'=y$ and $y'=x$ and, if I didn't make any mistakes, this reduces the problem to finding integers $x,y$ with $hx+ky=-1$ and $\tau, \tau'\in \mathbb H$ such that $$ \eta\left(\frac{x\tau+y}{k\tau-h}\right)\eta\left(\frac{y\tau'+x}{h\tau'-k}\right)=\eta(\tau)\eta(\tau')\sqrt{i(k\tau-h)(h\tau'-k)} $$ (I have taken exp again, but by (12) we do not lose information). Choosing $$ \tau'=\frac{x\tau+y}{k\tau-h} $$ leaves $$ \eta\left(\frac{\tau(yx+kx)+y^2-hx}{\tau(hx-k^2)+hy+hk}\right)=\eta(\tau)\sqrt{i(\tau(hx-k^2)+hy+hk)}. $$ When $hx-k^2>0$ we can apply the functional equation to the LHS to yield $$ \varepsilon(yx+kx, y^2-hx, hx-k^2, hy+hk)\sqrt{-i((hx-k^2)\tau+hy+hk)}\eta(\tau), $$ and thus it suffices to show $$ \varepsilon(yx+kx, y^2-hx, hx-k^2, hy+hk)=-i. $$ This is the same (by the definition of $\varepsilon$ and (12)) as showing that \begin{gather} \tag{1} \frac{yx+kx+hy+hk}{12(hx-k^2)}+s(-hy-hk, hx-k^2)=-\frac 12. \end{gather} In the case that $hx-k^2<0$ we can multiply all the entries of the "matrix" by $-1$ and the corresponding equation is \begin{gather} \tag{2} \frac{yx+kx+hy+hk}{12(hx-k^2)}+s(hy+hk, k^2-hx)=0. \end{gather}
Concluding remarks: For example, when $k=1$ we can choose $x=0$, $y=-1$ and equation (2) reads $0+s(0,1)=0$. The first non-trivial example is $k=2$, in that case $h$ must be odd and we can choose $x=1$, $y=-\frac{h+1}{2}$. When $h=1$ or $3$ we are in case (2) and one can verify it by hand. For $h>4$ we are in case (1) and we have the equation \begin{gather} \tag{3} -\frac{(h+1)(h-3)}{24(h-4)}+s\left(\frac{h^2-3h}{2}, h-4\right)=-\frac 12. \end{gather} For instance, when $h=5$ or $9$ the Dedekind sum is zero (for the latter case note that $h-4$ will divide $(h^2-3h)^2+4$) and the fraction is $-\frac 12$. It is not clear to me what is going on, even in this case. Apostol lists several congruences for the Dedekind sums (sections 3.6 and 3.8), but none seemed useful to me here.
