Questions tagged [theta-functions]
For questions about $\theta$ functions (special functions of several complex variables).
165 questions
3
votes
0answers
81 views
+100
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 ...
3
votes
0answers
22 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}...
0
votes
0answers
13 views
Finding doubly periodic solutions to partial differential equations
Say I have a certain PDE in real variables $x$ and $y,$ which might be nonlinear, so that we can't necessarily just throw a Fourier series at it.
By way of some intuition, let's say, I have a very ...
1
vote
1answer
40 views
Transformation of a theta function
Given $\Theta(\tau)=\sum_{n \in \mathbb Z}exp(2\pi in² \tau)$ and $\tau \in \mathbb H$
I am trying to prove the following identity:
$\Theta(-\frac{1}{2\pi})=\sqrt{\frac{\tau}{i}}\Theta(\frac{\tau}{2}...
8
votes
0answers
70 views
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 ...
1
vote
0answers
20 views
Evaluation of integral of multiple Jacobi theta functions
This problem is related to the Fermi gas problem of quantum mechanics.
Define $\rho(x,x')$ as follows
$$ \rho(x,x') = \theta_4(x) \frac{\theta_1^2(x/2)}{\theta_4^2(x/2)} \frac{\theta_2(\frac{x-x'}{2}...
4
votes
0answers
87 views
How many integer solutions are there to $x^4+y^4-x^2y^2=n$. Is there a generating function for this?
It would be kind of cool to get a closed form for the number of integer solutions $$x^4+y^4-x^2y^2=n$$ which we will let $\phi_n$ denote.
It would be cool because we could exploit $\sum_{n=1}^N\...
0
votes
0answers
36 views
Theta function equation
I have been trying to prove an equation of a theta function.
I understood that it some how related to the Poisson summation formula, but no luck. Any help would be appreciated.
the exercise
0
votes
0answers
26 views
Explicit Values of the Jacobi Theta Function [duplicate]
The sum $\sum_{n=-\infty}^{\infty}\exp(-\pi n^2) $, or $\vartheta(0;i)$ (Jacobi Theta Function) has a closed form solution of $\frac{\pi^{\frac{1}{4}}}{\Gamma(\frac{3}{4})}$ but nowhere have I been ...
2
votes
2answers
65 views
Evaluation of the limit $\lim_{q\rightarrow 1} \frac{\phi^5(q)_{\infty}}{\phi(q^5)_{\infty}}$
Given the Euler function $\phi(q)=\prod_{n = 1}^{\infty}(1-q^{n})$ which is a modular form where $q=\exp(2\pi i \tau)$, $|q|\lt1$
Then what is the limit $\lim_{q\rightarrow 1}\frac{\phi^5(q)_{\...
0
votes
0answers
26 views
how to derive the formula of Jacobi theta function which is below
What is the derivative of Jacobi theta function which is:
$$\Theta_3(z;\tau)= \sum_{n=-\infty}^\infty \exp(i\pi\tau n^2)\exp(2ni z)$$
and find its zeros.
0
votes
0answers
30 views
Analytic continuation of Riemann Theta function to positive semidefinite matrices
The Riemann $\Theta$-function is defined as
$$Θ(z|Ω)=\sum\limits_{q∈Z^N}e^{πiq⋅Ωq+2πiq⋅z},$$
where $\Omega$ has positive definite imaginary part to ensure convergence. In a particle physics ...
6
votes
2answers
220 views
How are the Jacobi Theta Functions analogous to the Exponential Function?
On the Wolfram MathWorld page on Jacobi Theta Functions, it says that the Theta Functions are elliptic analogues of the exponential function. Is this because they satisfy certain properties that the ...
4
votes
1answer
157 views
What's the connection between $\theta$ series and the number of integer solutions on a curve? Proof Verification
I just started learning about theta series and am now flexing my muscles. I hope that everything looks good. This is a request for a proof verification. I am asking for what the proper name in the ...
3
votes
1answer
181 views
How many integer pairs satisfy the ellipse $x^2+ay^2=r?$
How many integer pairs satisfy the ellipse $x^2+ay^2=r?$
What I have discovered thus far:
This post is largely to document the thinking that I have already done... I know that this can be frowned ...
2
votes
0answers
52 views
Clarification of Landau's Mechanics, Chapter.6 & Whittaker, Analytical Dynamics Chapter.6 needed
I am currently working through Landau's mechanics book and I am struggling to get my head around a solution provided in Chapter.6 relating to the asymmetrical top.
I fully comprehend how Landau ...
3
votes
1answer
120 views
How many integer solutions are there on an $n$ dimensional hypersphere of radius $\sqrt{r}$ centered at the origin?
Let $\phi(n,r)$ be the number of integer solutions of
$\sum\limits_{i=1}^n x_i^2=r$.
Then $\phi(2,r)=4\sum\limits_{d|r}\chi(d)$ where $\chi (x)=sin(\frac{\pi x}{2})=\cases{ 1\text{ when }x\cong 1 \...
0
votes
2answers
62 views
$\sum\limits_{\mathbb{d|n}}{f(d)}=\sum\limits_{\mathbb{d|n}}{g(d)}\implies f(n)=g(n)?$
Question:
Is it true that if for functions $f,g$ which map naturals to naturals
For all natural numbers n, we have $f(n)=g(n) \iff$ for all natural
numbers n we have
$\sum\limits_{\mathbb{d|...
2
votes
1answer
67 views
Properties of $\frac {\theta_{1}''(z|\tau)}{\theta_{1}(z|\tau)}$?
I'm looking for references concerning the properties of the function $\frac {\theta_{1}''(z|\tau)}{\theta_{1}(z|\tau)}$ where $\theta_{1}(z|\tau)$ is a Jacobi theta function defined here. I am trying ...
1
vote
0answers
42 views
$\Theta$ function in terms of Weierstraß $\sigma$ function?
Let $\Theta$ function be the function associated to a lattice $\Lambda=\oplus_{i\leq 2}Z\lambda_i\subset C$ of $C$ with transformation property defined as $\lambda\in\Lambda, \Theta(z+\lambda)=\Theta(...
2
votes
1answer
50 views
$\theta(z) = \sum_{n=-\infty}^{+\infty} e^{\frac{-n^2}{2}}e^{inz}$ How to show $\theta '(\frac{i}{2}) = \frac{-i}{2} \theta (\frac{i}{2})$?
$$\theta(z) = \sum_{n=-\infty}^{+\infty} e^{\frac{-n^2}{2}}e^{inz}$$
How to show $\theta '(\frac{i}{2}) = \frac{i}{2} \theta (\frac{i}{2})$?
0
votes
0answers
33 views
Hecke Operators and Eigenfunctions, Fourier coefficients
The problem statement, all variables and given/known data
Consider the action of $T_2$ acting on $M_k(\Gamma_{0}(N)) $, and show that $\theta^4(n)+16F $ and $F(t)$ are both eigenfunctions.
Functions ...
0
votes
0answers
51 views
How to show that the theta function is smooth?
I was reading on the theta function
$$
\vartheta(x) = \sum_{n=-\infty}^\infty e^{-\pi n^2 x}\qquad x>0
$$
The author claims that $\vartheta$ is smooth. Is there an easy way to show that this is ...
0
votes
0answers
13 views
Product of multi-dimensional theta functions
I am interested into the following sum of products of multi-dimensional theta functions on a genus-$g$ Riemann surface:
$$
\sum_{\ell}
\vartheta\!\!\begin{bmatrix} a + \frac{\ell}{k} \\ b_1 \end{...
0
votes
1answer
40 views
Basic $\theta$-function identity proof
For the $\theta$-function $$\theta (z) = \sum_{n \in \mathbb{z}} q^{n^2}e^{2\pi inz},$$
for $q$ given by $e^{\pi i\tau}$ for some $\tau \in \mathbb{C}$ with $Im(\tau) > 0$, suppose we've proved ...
18
votes
0answers
289 views
Asymptotic equivalent of $\sum_{n\ge0} q^{n^2}{x^n}$ as $x\to+\infty$
Let $q\in\Bbb C^*$ with $|q|<1$, define
$$f:x\mapsto\sum_{n\ge0} q^{n^2}{x^n}$$
I want to find an asymptotic equivalent of $f$ as $x\to+\infty$.
I found that
$$a\le|f(x)|\cdot\exp\left(\frac{\...
4
votes
1answer
64 views
The even-index reciprocal Lucas constant and $\sum_{n=1}^\infty \frac1{x_1^{2n}+x_2^{2n}}$
The sum of reciprocals of even index Lucas numbers has a nice closed-form in terms of theta functions,
$$\begin{aligned}S_e
&= \sum_{n=1}^\infty \frac1{L_{2n}}\\
&= \sum_{n=1}^\infty \frac1{\...
1
vote
0answers
38 views
proving sigma = BigTheta (BigΘ)
I'm trying to solve a BigΘ problem, and could use a little guidance to make sure I'm on the right path.
So my question is to show that $\sum_{i=1}^{n} i^{15} = Θ(n^{16})$
I know that for something ...
0
votes
0answers
36 views
Borel -summation for Riemann-Siegel theta function
Riemann-Siegel theta function is defined in terms of the Gamma function as: $\theta(t)=\arg(\Gamma(\frac{1+2it}{4}))-\frac{\log \pi}{2}t $ , it has asymptotic series which is not converge and it's ...
18
votes
2answers
372 views
An interesting identity involving Jacobi $\theta_4$ and $\zeta(2)$
A recent question mentioned an integral identity involving Dedekind $\eta$ function and a special value for the complete elliptic integral of the first kind. I refrained from providing a complete ...
6
votes
2answers
236 views
How to evaluate sums in the form $\sum_{k=-\infty}^\infty e^{-\pi n k^2}$
Online, one may find the values of the following sums:
$$\sum_{k=-\infty}^\infty e^{-\pi k^2}=\frac{\pi^{1/4}}{\Gamma(3/4)}$$
$$\sum_{k=-\infty}^\infty e^{-2\pi k^2}=\frac{\pi^{1/4}(6+4\sqrt 2)^{1/4}}{...
4
votes
0answers
107 views
Evaluate $ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty $
This identity is taken from a physics paper [1] stated without proof, on page 43.
$$ \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
=...
5
votes
1answer
148 views
Did Gauss know Jacobi's four squares theorem?
This is a question that i have already asked on HSM stackexchange, and i decided to ask it again here because it's more mathematical than historic (to make a conclusion in this question one needs more ...
0
votes
1answer
56 views
Linearly equivalent divisors induce same projective embedding
Consider the torus $T=\mathbb C^n/\Lambda$ for some lattice $\Lambda$.
Say that two divisors $A,B$ on $T$ are linearly equivalent if their difference is the divisor of a meromorphic function on $T$. ...
1
vote
0answers
56 views
Can we extend the theta function $\theta(z)$ to p-adic numbers $\mathbb{Z}_p$?
Let $\theta(z) = \sum_{n \in \mathbb{Z}} q^{n^2}$ with $q = e^{2\pi i n z}$. Can we extend the theta function to $p$-adic arguments? Here's an example:
$$ \theta( 1 + p^k) = \sum_{n \in \mathbb{Z}^...
4
votes
0answers
49 views
Has the idea of a 'hyperbolic Theta function' been studied?
Consider a uniform $\{m,n\}$ tiling of the hyperbolic plane, for convenience with one vertex at the origin (and also for convenience, normalize the edges to have unit hyperbolic length). Then there ...
1
vote
0answers
46 views
No weight $2$ newforms on $\Gamma_0(4)$?
How could it be there are no modular forms of weight $2$ and level $4$. In the lmfdb database? We can easily think of one: $\theta(z)^4$ with
$\theta(z)=\sum_{n\in \mathbb{Z}} q^{n^2}$ . Where are ...
2
votes
1answer
70 views
$ \sum_{n=1}^{\infty} \frac{ x^{n^2} (1 + x^n) - x^n}{1 -x^n} = 0.$ ??
While studying theta functions I noticed
$$ \sum_{n=1}^{\infty} \frac{ x^{n^2} (1 + x^n) - x^n}{1 -x^n} = 0.$$
Why is that so ??
Is there a similar case with a term $x^{n^3}$ ??
1
vote
0answers
29 views
Theta notation for Strassen's multiplication
A student discovers a way to multiply 2×2 matrices using exactly 5 multiplications, instead of Strassen’s 7. What is the number M(n) of multiplications for the resulting algorithm to multiply n×n ...
2
votes
1answer
123 views
Representation of integers as quadratic forms with integer coefficients
While reading the book The sensual (quadratic) form by J.H. Conway I got curious in this question. Maybe it is trivial, but I don't know how to answer it.
Let $f(x,y)=ax^2+hxy+by^2$, $g(x,y)=a'x^2+h'...
2
votes
1answer
266 views
Infinite series containing ceiling function
I am looking for a closed-form solution to this infinite (converging) series containing a ceiling function:
\begin{equation}\sum_{k=1}^{\infty}\frac{1}{2^{2k+\lceil 2\sqrt{k} \rceil}} \approx 0....
3
votes
2answers
110 views
How can we show that $R=S?$
Let
$$R=\prod_{n=1}^{\infty}\left({q^{2n-1}-1\over q^{2n-1}+1}\right)^2
\left({q^{2n}-1\over q^{2n}+1}\right)^2\tag1$$
and
$$S=2\prod_{n=1}^{\infty}\left({q^{2n}+1\over q^{2n}-1}\right)^{2(-1)^{n+1}}...
0
votes
1answer
68 views
Do these summations satisfy modular properties
In this post ,I observed computationally that the mock theta functions of order $3$,found in this wikipedia article
$f(q)=\sum_{n=0}^{\infty} \frac{q^{n^2}}{(-q;q)^2_{n}}$,$\phi(q)=\sum_{n=0}^{\infty}...
1
vote
1answer
213 views
Modular transformation in terms of generators
I was looking for a "general" modular transformation for the first kind of Jacobi theta function, $\theta_1(u,\tau)$. I do know how this theta function transforms under the two generators of modular ...
1
vote
1answer
130 views
conjectured relations of mock theta functions of order $3$
Given the following mock theta functions of order $3$,found in this wikipedia article
$f(q)=\sum_{n=0}^{\infty} \frac{q^{n^2}}{(-q;q)^2_{n}}$,$\phi(q)=\sum_{n=0}^{\infty} \frac{q^{n^2}}{(-q^2;q^2)_{n}...
2
votes
0answers
133 views
A more general closed-form of an integral involving a square power of $\theta_4$ - function
$\textbf{Problem statement}$. Inspired by the computations at this nospoon
we introduce the following integral:
$$\int_{0}^{\infty }\frac{~\theta _{4}^{2}\left( \exp \left( -\pi \,y\,\beta
\right) \...
4
votes
0answers
79 views
Reciprocal sums
How can I prove the following?
1)$$\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}^{...
2
votes
1answer
185 views
Applications of Jacobi theta functions [closed]
I've heard that the Jacobi theta function has wide applications, for instance in physics and proving elementary facts about the Fibonnaci sequence.
I'd be interested in knowing some of the most ...
4
votes
0answers
59 views
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 &...
2
votes
0answers
49 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 ...
