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What is the early history of the Dedekind eta function? In particular, why is it called the Dedekind eta function? Ramanujan's paper on the 24th power of the Dedekind eta function had appeared by 1916 and I have a reasonably good understanding what happened after that, in the other direction I know that by 1775 Euler had consider the Dedekind eta function divided by q1/24 in his work which led to the pentagonal number theorem, but what happened between Euler and Ramanujan that led to the introduction of the factor of q1/24 (which gives a half integer weight modular form) and the attachment of Dedekind's name to it?

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I believe that Dedekind introduced this function in Schreiben an Herrn Borchardt über die Theorie der elliptischen Modulfunktionen which was published in Crelle's Journal in 1877. The complete references is Journal für reine und angewandte Mathematik, 83 (1877) 265-292 and it is reprinted in volume 1 of his collected works with a postscript by Fricke explaining that this work remained largely unknown for some time. He also cites a later paper by F. Klein with substantial overlap.

The relevant equation in the paper is equation (24). Right after it he bemoans his lack of luck in finding the explicit expression for the eta function from the definition or without the use of theta functions:

Allein es ist mir bisher nicht geglückt, diese Darstellung von $\eta(\omega)$ als explizite Funktion von $\omega$ lediglich aus ihrer obigen Definition, also ohne die Hilfe der Theorie der $\vartheta$-Funktionen abzuleiten.

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  • $\begingroup$ Interesting, this is different from (and possibly earlier than?) the Dedekind reference that I found. Thanks! $\endgroup$ Dec 6, 2009 at 16:59
  • $\begingroup$ The paper you found might indeed have been written earlier. It was published later because, as you point out, it appeared with Riemann's collected works in 1892, but in Dedekind's collected works it appears earlier than the one I quoted. In fact, the one I quoted is paper XIV and the Erläuterungen... is paper XIII. $\endgroup$ Dec 6, 2009 at 17:17
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After some searching I found some relevant references:

H. Rademacher and A. Whiteman, “Theorems on Dedekind sums,” Amer. J. Math. 63 (1941), 377–407.

Arias-de-Reyna, J. Riemann's fragment on limit values of elliptic modular functions. Ramanujan J. 8 (2004), no. 1, 57--123.

Dedekind's paper is titled "Erlauterungen zu den vorstehenden Fragmenten" and was published in Riemann's collected works. In this paper Dedekind attempts to elucidate Riemann's “Fragmente uber die Grenzfalle der Elliptischen Modulfunctionen,”

It looks like Dedekind's introduction of the eta function was inspired by these fragmentary notes of Riemann. It would take me some time to parse the relevant issues (e.g. to what extent did Riemann know the Dedekind eta function, how might things have seemed to Dedekind, etc.) and I will not do so at the moment, but am happy to have found these for future reference.

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Perhaps Vladut's book gives infos on that, the old Enzyklopädie der mathematischen Wissenschaften usually contain detailed infos, then there are Klein's history books and Klein/Fricke's 2 vol.s on modular functions.

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I believe Dedekind introduced in the late 19th century in relation to elliptic functions and expressed them in terms of what is now known as Dedekind sums. Even though both the Euler function and the eta function are symmetric on the modular group, they may have been studied in different contexts where Dedekind's would be considered more directly related to modular forms.

In the pentagonal number theorem, there's no assumption or use of $x = e^{2\pi i z}$ in his function $\prod_n (1-x^n)$, and maybe it was not considered in relation to elliptic functions or modular forms at the time. Of course, the generalization of the pentagonal number theorem is to the triple product formula which is very much related to theta functions and modular forms as well.

If you notice on the Wikipedia article for the Euler function, they mention the two being related by a "Ramanujan's identity."

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I suspect the introduction of the factor $q^{\frac{1}{24}}$ into what is now called "Dedekind eta function" has even earlier origins than in Riemann's unpublished fragments. In his unpublished manuscripts on elliptic functions and modular forms, Gauss wrote down a multitude of connections between Euler function $\phi(q)=\prod_{k=1}^\infty (1-q^k)$ and Jacobi theta functions, and there are several appearances of the factor $q^{\frac{1}{24}}$ in his writings. In one of those appearances, Gauss even emphasizes the importance of Dedekind's modular form by referring to a certain result involving it as "one of the most important theorems of the theory", and gives the value of $\eta(i)$ as $0.768225$ (Gauss's werke, volume 3, p. 441).

This (previously mentioned) "most important theorem" of Gauss is equivalent to the following statement, which relates any three values $\eta(\tau),\eta(2\tau),\eta(4\tau)$ by the following equation:

$$\eta(2\tau)^{24} = \eta(\tau)^{16}\eta(4\tau)^8+16\eta(\tau)^8\eta(4\tau)^{16}$$

so expressing $\eta(4\tau)^8$ in terms of $\eta(\tau)$ and $\eta(2\tau)$ reduces into solving a simple quadratic equation.

In another place (Gauss's werke, volume 3, p. 456) Gauss writes:

$$[x] = \frac{a^{\frac{1}{12}}b^{\frac{1}{3}}(a^2-b^2)^{\frac{1}{24}}}{2^{\frac{1}{3}}x^{\frac{1}{24}}\sqrt{h}}$$

here $[x]=\prod_{k=1}^\infty (1-x^k)$ is Gauss's notation for Euler's function, and: $a=h\cdot\vartheta_3^2(x), b = h\cdot\vartheta_4^2(x)$. Rewriting it as:

$$x^{\frac{1}{24}}[x] = \frac{a^{\frac{1}{12}}b^{\frac{1}{3}}(a^2-b^2)^{\frac{1}{24}}}{2^{\frac{1}{3}}\sqrt{h}}$$

one sees that this is actually an expression for Dedekind eta function with $q$ denoted as $x$. Actually I'm not quite sure that this result is exact - I compared it with known results about the relation of Dedekind eta function to Jacobi theta function and everything agrees except the constant factor $2^{-\frac{1}{3}}$ - it should be $2^{-\frac{1}{6}}$.

I stress those facts to show that even in Gauss's writings, the $24$th power (an important integer occurring also in other mathematical fields) appears, although I'm still not sure of the exact correctness of several of Gauss's results involving it.

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