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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:

  1. Why is the second transformation formula better than the first one?
  2. Why are Dedekind sums interesting?
  3. Is there a more abstract approach to the transformations of the Dedekind eta function?
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    $\begingroup$ I recommend this article by Étienne Ghys in the Proceedings of ICM 2006: icm2006.org/proceedings/Vol_I/15.pdf (sections 3.2 to 3.5). You will find for example a relation between the Rademacher function (which is essentially the Dedekind sum) and the linking number between two knots. There are also references to many other connections (e.g. in Atiyah's article). $\endgroup$ – François Brunault Apr 15 at 22:19
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    $\begingroup$ Dedekind sums (and some generalizations) appear naturally, for example, in special values of zeta functions associated to totally real number fields. Zagier has a nice article on this. See also the article by Gunnells and Sczech. $\endgroup$ – EFinat-S Apr 16 at 0:18
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    $\begingroup$ The 2nd transformation is better than the first because it introduces us to Dedekind sums, which are interesting. What's interesting about Dedekind sums can be foound in the book, Rademacher & Grosswald, Dedekind Sums, ams.org/books/car/016/car016-endmatter.pdf Among other things, they turn up in the analysis of linear congruential pseudorandom number generators. $\endgroup$ – Gerry Myerson Apr 16 at 0:24
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    $\begingroup$ The second formula is more naturally stated for $\log \eta(\tau)$, it gives more information in the sense that it gives a natural lift to $\frac{2\pi i}{24} \mathbb{Z}$ of the $24$-th root of unity $\varepsilon$. This is the Rademacher function which appears in many different areas. On the other hand, from a purely computational point of view, the first formula is of course faster. It is also better suited to find a criterion for when a product of eta functions is modular for some congruence subgroup, see Newman, Construction and application of a class of modular functions II. $\endgroup$ – François Brunault Apr 16 at 9:35

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