Questions tagged [theta-functions]

For questions about $\theta$ functions (special functions of several complex variables).

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Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
Pascal's user avatar
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11 votes
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Global sections of vector bundles on a complex elliptic curve and analytic functions

Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I ...
Alex's user avatar
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9 votes
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How to estimate a specific infinite sum

Let $M$ be an $n$ by $n$ matrix with each diagonal element equal to $k$ and each non-diagonal element equal to $k-1$ where $n$ and $k$ are positive integers. Let $k < n$ and we can assume both $k$ ...
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9 votes
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How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$

I'd like to simplify $$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form $$...
Neves's user avatar
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A relation concerning the "sum of squares" counting function $r_2(n)$

Let $r_2(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here the sign as well as order of summands matters. Also by convention we ...
Paramanand Singh's user avatar
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8 votes
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Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$ \theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) , $$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
E.P.'s user avatar
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7 votes
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Primes of the form $x^2 + n y^2$ using theta function

It is well known that if $p$ is a prime, then $p$ can be written in the form $x^2 + n y^2$ under certain congruences conditions. For example, \begin{align} p = x^2+y^2 &\Leftrightarrow p = 2 \text{...
Zakhurf's user avatar
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On Jacobi theta functions and some curious evaluations

(This post has been modified to include $\vartheta_2$.) The Jacobi theta functions are well-known for the identity, $$\vartheta_2^4 - \vartheta_3^4 +\vartheta_4^4 = 0 $$ so this post will involve 4th ...
Tito Piezas III's user avatar
6 votes
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413 views

Jacobi Theta Function on the Unit Circle - Is there a Limit in the Distribution Sense?

The third Jacobi theta function $$\theta_{3}\left(z,q\right)=1+2\sum_{n=1}^{\infty}q^{n^{2}}\cos\left(2\pi n z\right)$$ appears in the study of path integrals in QM. Specifically in the problem of a ...
eranreches's user avatar
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Calculating $\Theta$ series of $E_8$ Lattice

I'm trying to calculate the $\Theta$ series of $E_8$ lattice, using the following Gram matrix (the Cartan Matrix of $E_8$): $$\left(\begin{matrix} 2 & 0 & 0 & -1 & 0 & 0 & 0 &...
Xiang Zhao's user avatar
6 votes
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379 views

Evaluating the alternating hyperbolic series $\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k \sinh (\pi k)}$

You can evaluate the alternating hyperbolic series $$\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{2n-1} \sinh (\pi k)}$$ for any positive even value of $n$ by integrating the function $$\frac{\pi \csc (\pi ...
Random Variable's user avatar
6 votes
2 answers
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Jacobi Theta Functions?

For the Jacobi theta function $\vartheta_3(z|\tau)$ there exists an equality (by Whittaker & Watson) \begin{equation} \vartheta_3(z|\tau) = \sum_{n=-\infty}^{\infty} e^{n^2 \pi i \tau + 2 n i z} =...
Erich's user avatar
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Is this function an alternative solution to the nonlinear pendulum?

Is this function an alternative solution to the nonlinear pendulum? Introduction I am working with the differential equation of the frictionless nonlinear pendulum: $$\ddot{\theta}(t) + b\,\sin(\theta(...
Joako's user avatar
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Validity of $\ln z=\frac{\pi}{\operatorname{AGP}(\theta_2^2(1/z),\theta_3^2(1/z))}$

Definitions For the definitions of $\operatorname{AGP}$ and $\operatorname{AGO}$, see here or here. $\theta_2(z)$ and $\theta_3(z)$ are defined as follows: $$\theta_2(z)=\sum_{n=-\infty}^\infty z^{(n+...
Wane's user avatar
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Abel's summation formula and approximating an integral of Jacobi theta functions

As my previous post remains unanswered, I thought I would post a more complete form of the problem in case it would be more practical to work on/ solve. I am trying to compute the following integral \...
Saïd M's user avatar
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Has the idea of a 'hyperbolic Theta function' been studied?

Consider a uniform $\{m,n\}$ tiling of the hyperbolic plane, for convenience with one vertex at the origin (and also for convenience, normalize the edges to have unit hyperbolic length). Then there ...
Steven Stadnicki's user avatar
4 votes
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What is the coefficient of $q^n$ in the Lambert Series Expansion $a^2(q).a(q^5)$

What is the coefficient of $q^n$ in the Lambert Series Expansion $a^2(q).a(q^5)$. Here $a(q)$ is Borwein Theta Function. I am using Ramanujan's Theta Functions Book by Shaun Cooper as a reference ...
Dumbledory's user avatar
4 votes
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The Four Square Theorem and Integral Apollonian Circle Packings, is there any connection?

I have been studying theta-functions and made an interesting observation which I have a question about QUESTION: Is there a more intuitive, in particular a mostly geometric way, to prove the four ...
Matt Calhoun's user avatar
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303 views

Relationship between $\theta$ functions and number fields.

I'm trying to have a clear picture of the relationship of theta functions and $L$-functions, and the geometric objects they relate to. Firstly, I know that $\theta$-functions arise as sections of ...
Elmoco's user avatar
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Closed form of $\sum_{n=1}^\infty q^{- n^2} z^n$

In this question the summation goes from $-\infty$ to $\infty$ and the answer has a pretty "good" closed form. Now I came across the sum $\sum_{n=1}^\infty q^{-n^2} z^n$ where $|z|<1$ and I don't ...
Mikalai Parshutsich's user avatar
4 votes
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136 views

How many integer solutions are there to $x^4+y^4-x^2y^2=n$. Is there a generating function for this?

It would be kind of cool to get a closed form for the number of integer solutions $$x^4+y^4-x^2y^2=n$$ which we will let $\phi_n$ denote. It would be cool because we could exploit $\sum_{n=1}^N\...
Mason's user avatar
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Evaluate $ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty $

This identity is taken from a physics paper [1] stated without proof, on page 43. $$ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty =...
cactus314's user avatar
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4 votes
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98 views

Reciprocal sums

How can I prove the following? 1)$$\sum_{k=0}^{\infty}\frac{1}{F_{2k+1}}=\frac{\sqrt{5}}{4}\theta_{2}^{2}(0,\frac{3-\sqrt{5}}{2}),$$ 2)$$\sum_{k=0}^{\infty}\frac{1}{F_{k}^{2}}=\frac{5}{24}(\theta_{2}^{...
user423822's user avatar
4 votes
1 answer
268 views

About the sums $\sum_{n=1}^\infty x^{n^2}$ and $\sum_{n=1}^\infty \frac{x^n}{1+x^{2n}}$

Despite all my efforts trying to crack these, i haven't been able to do so. An approach that i've tried gives me somewhat of an asymptotic approximation, but still fails to produce the values near x=0....
Rafa's user avatar
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0 answers
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Is $\sum_n\exp(ian+ibn^2+icn^3)$ known in terms of anything else?

For arbitrary $a,b,c$, does the series $$F(a,b,c)=\sum_{n=-\infty}^\infty\exp\left(ian+ibn^2+icn^3\right),$$ i.e. an evenly-weighed series of exponentials of cubic polynomials, converge to anything ...
E.P.'s user avatar
  • 2,481
4 votes
1 answer
115 views

Injectivity of theta function

Let $\vartheta_{00}$ and $\vartheta_{01}$ be Jacobian theta functions (notations like on wikipedia). $$F:=\left\{ \tau \in \mathbb{C}: \mathrm{Im}(\tau)>0, \left| \mathrm{Re}(\tau)\right|<1, \...
linksideal's user avatar
4 votes
0 answers
442 views

Modular forms on the theta group

The theta group $\Gamma$ is the subgroup of $SL(2;\mathbb{Z})$ generated by $T=\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.$ It ...
user164947's user avatar
3 votes
0 answers
136 views

Integral $\int_0^1\ln^2(t)\psi^8(-t)dt=\frac{7}{8}\zeta(3)$

I was working on the following Integral: $$I=\int_0^1\ln^2(t)\psi^8(-t)dt=\frac{7}{8}\zeta(3)$$ Where, $$\psi(q)=\sum_{n=-\infty}^{\infty}q^{n(2n+1)}$$ is a Ramanujan Theta Function. Now it is well ...
Miracle Invoker's user avatar
3 votes
0 answers
129 views

How did people come up with the product formula for Jacobi's theta function

I understand that the function $\Theta(z|\tau)$, with its definition $$\Theta(z | \tau) := \displaystyle\sum_{n = -\infty}^{\infty} e^{\pi in^2\tau} e^{2\pi inz},$$ satisfies the product formula $$\...
Squirrel-Power's user avatar
3 votes
0 answers
29 views

$(\mathfrak{g}, K)$-module for covering group of $\mathrm{GL}_2(\mathbb{R})$

Harich-Chandra introduced the notion of $(\mathfrak{g}, K)$-module to study representations of a real Lie group $G$ via its Lie algebra $\mathfrak{g}$ and a maximal compact subgroup $K$. One also has ...
Seewoo Lee's user avatar
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3 votes
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99 views

Line bundle of complex tori

Let $\Lambda \subset V=\mathbb{C}^g$ be a lattice and $T_g=V/\Lambda$ be a $g$ dimensional complex torus. According to the Appell-Humbert theorem, any line bundle $L$ of $T_g$ is isomorphic to the ...
user682141's user avatar
3 votes
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74 views

A "surprising" asymptotic inverse of $\vartheta _3(0,x)$

After this question of mine related to the problem of approximate solutions of $$\large\color{red}{\operatorname{\vartheta}_3}\left(0,x\right)=k$$ when $k$ is large, continuing the previous work (just ...
Claude Leibovici's user avatar
3 votes
0 answers
80 views

From root and weight lattices of SU(N) to $\theta$-functions as sections of a line bundle and $CP$-space

I have troubles to digest the following messages/discussions in the following work in p.10-12; Which construct a map from the moduli space of flat connections $M_{\rm flat}=\mathbb{E} / {\mathfrak S}...
annie marie cœur's user avatar
3 votes
0 answers
164 views

Discrete theta functions and periodicity

I'm doing quantum mechanics and I have an eigenfunction which is a theta function. I then discretised it, since I want see if I can find the eigenvalues for the discrete case by finding the ...
Lewis Proctor's user avatar
3 votes
0 answers
566 views

Definite integral of Jacobi Theta Function

I need to evaluate an integral that involves the Jacobi Theta Function $\vartheta_3(z,q)$ defined by MathWorld as $$\vartheta_3(z,q)=\sum_{n\in\mathbb{Z}}q^{n^2}e^{2inz}.$$ Specifically, I wish to ...
goodwitm's user avatar
3 votes
0 answers
839 views

Implementation of Jacobi theta functions in Matlab

I need Jacobi theta functions for my Matlab program. The functions are not included in the predefined Matlab functions. Doing a simple Google search, I found a package developed by Moiseev I. in 2008. ...
lina's user avatar
  • 31
3 votes
0 answers
124 views

Degree of the Divisor of a Theta Function

Let $(\gamma_1, \gamma_2)$ be a base for a lattice $\Gamma$ in $\mathbb C$, and $\theta$ a theta function, ie an holomorfic function such that $\theta(z+ \gamma) = \theta(z)e^{2i\pi(a_\gamma z + b_{\...
Maffred's user avatar
  • 4,016
3 votes
0 answers
98 views

What is known about $\sum_{n=0}^{\infty} x^{n^3} $.

$f(x) =\sum_{n=0}^{\infty} x^{n^2}$ and similar "theta-type" functions are extensively studied. They have many properties and occur in number theory , algebra (in particular solving the quintic ...
mick's user avatar
  • 16k
3 votes
0 answers
443 views

Values of derivatives of Jacobi theta function

The Jacobi theta function is defined as: $$\theta(x)=\sum_{n=-\infty}^{\infty}\exp(-\pi n^2 x)\text{ }, x>0$$ On wikipedia.org, we can find close-form expressions for the values of $\theta(k)$,$...
mike's user avatar
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3 votes
0 answers
285 views

A Dedekind eta function sum of form $y_0^k+y_1^k+y_2^k+y_3^k+... = 0$

Given the Dedekind eta function $\eta(\tau)$. Define, $y_p = e^{\pi i p/6}\,\eta(\tfrac{\tau+2p}{5})$ Prove the multi-grade identity [1], $y_0^k + y_1^k + y_2^k + y_3^k + y_4^k + (\sqrt{5}\,\eta(5\...
Tito Piezas III's user avatar
2 votes
0 answers
36 views

How does Jacobi form lives projectively on the torus $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$?

I saw the statement in the question from the book Moonshine Beyond the Monster. We are given the definition : a group hom $\rho : G \rightarrow PGL(V)$ is called a projective representation. I can't ...
Mahammad Yusifov's user avatar
2 votes
0 answers
32 views

Elliptic functions by Eisenstein-Kronecker

In $\textit{Elliptic Fuctions according to Eisenstein and Kronecker}$, chap VIII, section 13 by A.Weil there is the following problem For any integer $k \geq 0$ and $z, w \in \mathbb{C}$, the function ...
Mario's user avatar
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2 votes
0 answers
79 views

Functional equation for $\theta$ function using functional equation of $\zeta(s)$

Let $$\theta(t) = \sum_{n \in \mathbb Z}e^{-n^2 \pi t}.$$ We can derive a functional equation for $\theta$ using Poisson summation formula: $$\theta(1/t) = \sqrt t \theta(t). $$ Riemann uses the above ...
Eloon_Mask_P's user avatar
2 votes
0 answers
59 views

The theta function on Weierstrass normal form

We want to write down the theta function of the point on Weierstrass normal form. Now, we can embed an elliptic curve to projective plane by$E=\mathbb{C}/(\tau\mathbb{Z}+\mathbb{Z})\to \mathbb{P}^2$ ...
Yos's user avatar
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2 votes
0 answers
78 views

A sum related to the theta function

I am interested in the serie $$\sum_{n=1}^\infty(2n+1)e^{-tn(n+1)}$$ for something I am working on. Wolfram alpha gets stuck on this problem, but it gives the following result for a closely related ...
Daniel Robert-Nicoud's user avatar
2 votes
0 answers
84 views

Questions about convergence related to Theta functions.

Let $H_n$ be the set of matrices $z\in \mathbb{C}^{n\times n}$ which are symmetric with a positiv-definite imaginary part. Let's assume that it is already known and proved that $\vartheta_a(z)=\sum_{g\...
Tom's user avatar
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2 votes
0 answers
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Adelic theta function over function fields

I saw the following claim on Godement-Jacquet's classical book "Zeta functions of simple algebras": (on page 153) Let $F$ be a global function field, $\Phi$ a Schwartz function on $\mathbb{A}...
youknowwho's user avatar
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2 votes
0 answers
51 views

Need help understanding proof of the functional equation for the theta function.

I am following the proof of theorem 1.3 above until at the point the author finds the recurrence relationship $a_n = a_{n-k}e^{b - 2\pi ni\tau}$. When $k = 0$ we have $a_n = a_{n}e^{b - 2\pi ni\tau} \...
Abel's user avatar
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2 votes
0 answers
109 views

Is there any definition for this series function $f(s)=\sum_{n=1}^\infty e^{-n^s}$?

What I am asking for is if there is any theory related to this real series $f(s)=\sum_{n=1}^\infty e^{-n^s}$ and $s\ge 1$. As far as I know, if $s = 1$, it's a simple geometric series, and if $s = 2$, ...
Eddie Lin's user avatar
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2 votes
0 answers
57 views

Weierstrass factorization of $\theta_1(z|\tau)$ in $z$

Suppose $f$ is an entire function and $f(0)\ne 0$. Let $z_1, z_2,\ldots$ be the zeros of $f$ and $p_1, p_2,\ldots$ be any sequence of nonnegative integers such that $$r\gt 0\implies \sum_{n=1}^\infty \...
1mik1's user avatar
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