The Dedekind function is defined as follows $$\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),\qquad q=e^{2\pi i\tau}.$$ We have $$\eta(\tau+1)=\zeta_{24}\eta(\tau),\qquad \eta\left(-\frac{1}{\tau}\right)=\sqrt{-i\tau}\eta(\tau).$$

For a general unimodular substitution ($c\geq 0$, $c=2^\lambda c_1$, $c_1$ odd, if $c=0$ then $c_1=1$)

$$\eta\left(\frac{a\tau+b}{c\tau+d}\right)=\left(\frac{a}{c_1}\right)\zeta_{24}^{(bd(1-c^2)+c(a+d)+3(1-c_1)+3a(c-c_1)+\lambda\frac{3}{2}(a^2-1))}\sqrt{-i(c\tau+d)}\eta(\tau).$$

But we also have

$$\eta\left(\frac{a\tau+b}{c\tau+d}\right)=\varepsilon(a,b,c,d)\sqrt{-i(c\tau+d)}\eta(\tau),$$ where $$\varepsilon(a,b,c,d)=\exp\left(\pi i\left(\frac{a+d}{12c}+s(-d,c)\right)\right),$$ $$s(h,k)=\sum_{r=1}^{k-1}\frac{r}{k}\left(\frac{hr}{k}-\bigg\lfloor\frac{hr}{k}\bigg\rfloor-\frac{1}{2}\right).$$ The sum $s(h,k)$ is the so called Dedekind sum. My questions are:

- Why is the second transformation formula better than the first one?
- Why are Dedekind sums interesting?
- Is there a more abstract approach to the transformations of the Dedekind eta function?