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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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:
$$-\int_1^{\infty } \left(t^{-s-...
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How to find the Lambert series expansion of this function
Let's consider the function
\begin{align}
\frac{\eta ^m\left( q \right)}{\eta \left( q^m \right)}
\end{align}
here η is the Dedekind eta function $
\eta \left( q \right) =q^{\frac{1}{24}}\prod_{n=1}^{\...
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how to evaluate the explicit formula for the quotient powers of the Dedekind eta function
I'm working on a thesis in number theory, specifically focusing on modular forms, particularly the Dedekind eta functions. I want to know if there is a way to obtain the explicit expression for the ...
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The Dedekind eta function $\eta(\tau)=q^{\frac{1}{24}} \prod_{n=1}^\infty (1-q^n)$ and $|\tau|^{1/2} |\eta(\tau)|^2$
I tried to prove the standard identities of the Dedekind eta function
$$\eta(\tau)=q^{\frac{1}{24}} \prod_{n=1}^\infty (1-q^n),$$
where $q=\exp(2\pi i \tau)$ for some complex number $\tau$, but ...
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For which of the Weierstrass elliptic function periods do this equation of the modular discriminant and the Dedekind eta function apply?
It is often claimed that the following equation holds for the modular discriminant $\Delta=g_2^3-27g_3^2$ of the Weierstrass elliptic funtion and the Dedekind eta ($\eta$) function for period ratio $\...
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Summiation of the greatest integer function with terms of dedekind sums
I try to sum the greatest integer function like：
$$\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{...
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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#...
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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 ...
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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 ...
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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 ...
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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)=\...
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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/~...
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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 ...
user775699
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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{\...
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