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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.

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how to evaluate the explicit formula for the quotient powers of the Dedekind eta function

I'm working on a thesis in number theory, specifically focusing on modular forms, particularly the Dedekind eta functions. I want to know if there is a way to obtain the explicit expression for the ...
Sofiane Abdelhamid's user avatar
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The Dedekind eta function $\eta(\tau)=q^{\frac{1}{24}} \prod_{n=1}^\infty (1-q^n)$ and $|\tau|^{1/2} |\eta(\tau)|^2$

I tried to prove the standard identities of the Dedekind eta function $$\eta(\tau)=q^{\frac{1}{24}} \prod_{n=1}^\infty (1-q^n),$$ where $q=\exp(2\pi i \tau)$ for some complex number $\tau$, but ...
ShoutOutAndCalculate's user avatar
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For which of the Weierstrass elliptic function periods do this equation of the modular discriminant and the Dedekind eta function apply?

It is often claimed that the following equation holds for the modular discriminant $\Delta=g_2^3-27g_3^2$ of the Weierstrass elliptic funtion and the Dedekind eta ($\eta$) function for period ratio $\...
Arvid Samuelsson's user avatar
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Summiation of the greatest integer function with terms of dedekind sums

I try to sum the greatest integer function like: $$\sum_{i,j=0}^{m-1}\left\lfloor\frac{in_1 +jn_2}{m}\right\rfloor$$ This can be solved by using Hermite formula : $$\sum_{j=0}^{m-1} \left\lfloor\frac{...
Stringer Fan's user avatar
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The error function for the numerical Dedekind eta function?

The Dedekind eta function \begin{equation} \eta(\tau)=e^{\frac{\pi i \tau}{12}} \prod_{n=1}^\infty (1-e^{2n\pi i \tau}) =q^{\frac{1}{24}} \prod_{n=1}^\infty (1-q^n) \end{equation} where the Eulear ...
ShoutOutAndCalculate's user avatar
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Is there a general theorem relating the Dedekind eta function to polynomial roots?

This question The radical solution of a solvable 17th degree equation and these wikipedia pages: https://en.wikipedia.org/wiki/Plastic_number#Number_theory https://en.wikipedia.org/wiki/Bring_radical#...
userrandrand's user avatar
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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 ...
Jabberwocky's user avatar
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Geometric interpretation of Dedekind sum?

Dedekind sum can be defined for a pair of coprime numbers $s(n,m)$. It appears also in the SL$(2,\mathbb{Z})$ transformation of Dedekind eta function. Is there any geometric intuition what it is ...
jtkw's user avatar
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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 ...
Mattia Coloma's user avatar
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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_{\...
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Dedekind eta function at cusp other than infinity

(Sorry for my poor english...) Let $\triangle(z)=q\prod_{n=1}^{\infty}(1-q^n)^{24}$ be a cusp form of weight $12$. Let $N$ be a positive integer, $\delta \mid N$ and $\triangle_{\delta}(z)=\...
ililiil's user avatar
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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 http://sites.science.oregonstate.edu/~...
Freddie's user avatar
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Doubt in proof of Siegel of Dedekind eta function in transformation $S=-\frac{1}{\tau} $

I am self studying analytic number theory from Tom M Apostol Modular functions and Dirichlet series in number theory and I am having a doubt in Theorem 3.1 . ( I have doubt only in highlighted part of ...
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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{\...
Diffycue's user avatar
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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 $\...
Mohammad Al Jamal's user avatar
2 votes
2 answers
479 views

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 $s \to i \infty$, ...
user3141592's user avatar
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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 ...
aleden's user avatar
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11 votes
1 answer
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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 ...
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13 votes
2 answers
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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 = ...
Steven's user avatar
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