Why is Dedekind sum? 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?

 A: I'll answer (2) and (3) together. To elaborate from the many suggestions in the comments, Dedekind sums arise in the transformation law of the Dedekind eta function, which are in turn related to periods of Eisenstein series. The latter are connected to special values of $L$-functions, so it is natural that Dedekind sums are used to parametrized these special values. More abstractly, R. Sczech showed that they can be used to construct group cocycles on GL(2), and more generally GL(n), and are expected to realize elements in the Eisenstein cohomology in the sense of Harder (one generally believes this, but it is not proved). As such, Sczech used these cocycles to give a new proof of the Klingen-Siegel rationality theorem for partial zeta functions of totally real fields. Note that the reciprocity law of Dedekind sums (and their various generalizations) can be traced back to this cocycle property.
Dedekind sums in fact satisfy certain integrality properties, which might be expected since they parametrize $L$-values. Recent work of Dasgupta and Charollois refined this construction of generalized Dedekind sums to construct $p$-adic $L$-functions, and have used this as an ingredient in their proof of the tame refinement of the Gross-Stark conjectures. (They also show that the cocycle is equal to a certain Shintani cocycle, up to a coboundary, which is constructed using the geometry of certain cones.)
Dedekind sums, as homomorphisms on SL(2,$\mathbb Z$), can be used to construct what is called the Dedekind-Rademacher homomorphism on the congruence subgroup $\Gamma_0(p)$. This particular map has been used by Mazur to study the Eisenstein ideal, Merel to construct Eisenstein cycles in the relative homology of the modular curve $X_0(p)$, and Darmon et al to construct rigid analytic cocycles on SL$(2,\mathbb Z[1/p])$ that relate to certain real-multiplication (RM) points that parallel the behaviour of singular moduli. So lots and lots of arithmetic applications!
In another direction, since Dedekind sums capture the transformation law of the Dedekind eta function, they can be understood to be a measure of the "error of modularity" in a certain sense. This paradigm (plus other ideas) has led Zagier to define new objects called quantum modular forms.
Dedekind sums have also an incarnation over imaginary quadratic fields, called elliptic Dedekind sums, where in this case the special values of the $L$-function they parametrize are related to the period of an elliptic curve . They also have generalizations to analogues of Eisenstein cocycles on GL(n), and in this case the $L$-functions they parametrize are degree $n$ extensions of the imaginary quadratic field.
