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Questions tagged [dedekind-eta-function]

Use this tag for questions about a particular function defined on the upper half-plane of complex numbers and that is a modular form of weight one-half.

For any complex number $\tau$ with Im $\tau > 0,$ let $q = e^{2\pi i \tau}.$ Then the Dedekind eta function is $$\eta(\tau ) = e^{\frac{\pi i\tau}{12}} \prod_{n=1}^{\infty} \left(1 -e ^{2 n \pi i \tau}\right) = q^{\frac1{24}} \prod_{n=1}^{\infty} \left(1 - q^n\right).$$

The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it.

Because the eta function is easy to compute numerically from power series, it is often helpful in computation to express other functions in terms of it when possible. Products and quotients of eta functions, called eta quotients, can be used to express a great variety of modular forms.

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Which role does the $\frac{1}{24}$ in the Dedekind $\eta$-function play?

The Dedekind $\eta$-function is defined as $$\eta(z) = q^{\frac{1}{24}} \prod_{n = 1}^\infty (1 - q^n)^{-1}$$ where $q = e^{2 \pi i z}$. My question is: If I start with the Euler-product $\prod_{n = 1}^\infty (1 - q^n)^{-1}$, how do I come to the…
Steven
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Modular transformations of $\eta(\tau)$

Under a modular transformation the Dedekind $\eta$ function transforms as $$\eta(-1/\tau) = \sqrt{-i \tau}\eta(\tau) \, .\tag*{$(*)$}$$Siegel gives a proof in this paper here that uses complex analytic techniques. However, I am wondering how we can…
user338358
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How to derive relationship between Dedekind's $\eta$ function and $\Gamma(\frac{1}{4})$

I am trying to determine in what way to approach finding a connection between Dedekind's Eta Function, defined as $$\eta(\tau)=q^\frac{1}{24}\prod_{n=1}^\infty(1-q^n)$$ where $q=e^{2\pi i \tau}$ is referred to as the nome. and the Gamma Function…
aleden
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The golden ratio $\phi$ for $_2F_1\big(\frac16,\frac16,\frac23,-2^7\phi^9\big)$ and $_2F_1\big(\frac16,\frac56,1,\frac{\phi^{-5}}{5\sqrt5}\big)$?

I. Context Given the golden ratio $\phi$, then we have the nice closed-forms, \begin{align} _2F_1\left(\frac16,\frac16,\frac23,-2^7\phi^9\right) &= \frac{3}{5^{5/6}}\phi^{-1}\\[6pt] _2F_1\left(\frac16,\frac56,1,\,\frac{\phi^{-5}}{5\sqrt5}\right)\,…
Tito Piezas III
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Proof of the transformation formula of the Dedekind eta function using the Euler-Maclaurin formula

Using the Euler-Maclaurin summation formula, we can prove that$$\log η(τ)=\frac{iπτ}{12}-\frac{iπ}{12τ}-\log\sqrt{-iτ}-\int_0^∞p(y)\left[\frac{2πiτ}{e^{-2πiτy}-1}+\frac1y\right]\,\mathrm dy,$$where $\eta(\tau)$ is the Dedekind eta function, and…
Mohammad Al Jamal
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Exercise 3.6 from Apostol's "Modular functions and Dirichlet series"

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…
Croqueta
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How to find the Lambert series expansion of this function

Let's consider the function \begin{align} \frac{\eta ^m\left( q \right)}{\eta \left( q^m \right)} \end{align} here η is the Dedekind eta function $ \eta \left( q \right) =q^{\frac{1}{24}}\prod_{n=1}^{\infty}{\left( 1-q^n \right)} $,m is a positive…
Loyar
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What is precisely the connection between the leech lattice and the Dedekind eta functions?

I've recently seen stated here https://en.wikipedia.org/wiki/Dedekind_eta_function#Definition, here https://math.stackexchange.com/a/1754273/917010, here: https://bahasa.wiki/nn/Dedekind_eta_function and in a couple more sites that there supposedly…
Jabberwocky
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Level $8$ modular form eta-quotient discrepancy - vanish of order $1/2$?

Something is wrong here. I have an eta-quotient $$g(z) := \eta^{2}(z)\eta(2z)\eta(4z)\eta^{2}(8z),$$ which belongs to $S_{3}(8, \chi)$ according to page 3 of…
Freddie
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Proof of a Dedekind eta-product identity of level 8 and degree 23.

Define the notation $\, u_n := \eta(n\tau) \,$ for the Dedekind $\eta$ function and $\, q := e^{2\pi i\tau}.$ The simplest identity of level $4$ and degree $24$ seems to be $$ 0 = u_1^{16}u_4^8 + 16 u_1^8 u_4^{16} - u_2^{24}. $$ The simplest…
Somos
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Zeta Function Zeros and Dedekind Eta Function Integral

Working on algorithms related to number theoretical function calculation performance improvement and accidentally discovered the following for Dedekind $\eta$ function: $$-\int_1^{\infty } \left(t^{-s-1}+t^{s-1}\right) \left(\frac{\pi t}{12}+\log…
Gevorg Hmayakyan
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Modularity of Euler $q$-series.

The Dedekind $\eta$ function is defined as a function on the upper half space $\mathbb{H}$ as $$\eta(\tau) = e^{\frac{\pi i \tau}{12}}\prod_{n>0}(1-e^{2\pi i n\tau})$$ or, using the circular variable $q=e^{2\pi i \tau}$, as the following…
Mattia Coloma
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Euler product exists for the Dedekind zeta function

The Dedekind zeta function of a number field $K$, denoted by $\zeta_K(s)$, is defined for all complex numbers $s$ with $\Re(s) > 1$ by the Dirichlet series \begin{equation*} \zeta_K(s) = \sum_{\mathfrak{a}}…
bozcan
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Closed form of $\eta^{(k)}(i)$.

Does anyone know closed form expressions for $$\eta^{(k)}(i)$$ up to high $k \in \mathbf{N}$? ($\eta$ is the Dedekind eta function.) For instance, I can use Mathematica to obtain $$\eta(i) = \frac{\Gamma \left(\frac{1}{4}\right)}{2 \pi…
Diffycue
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Location of the zeros of Dedekind Eta Function

Just a fast question, since I have not been able to find any answer for it online. Where are the zeros of Dedekind eta function $\eta(s)$ located? Apart from the trivial one as $s \to i \infty$, are there any other zeros in the upper half of the…
user3141592
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