# Questions tagged [theta-functions]

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94
questions

**8**

votes

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236 views

### Positive integer solutions of $ab+ac+ad+bc+bd+cd=n$

Consider a quadratic form $Q(a,b,c,d)=ab+ac+ad+bc+bd+cd$ on $\mathbb Z^4$.
For some reason I am interested in the number of solutions $(a,b,c,d)\in\mathbb Z_{> 0}^4$ of $Q(a,b,c,d)=n$ as a function ...

**1**

vote

**0**answers

77 views

### From a factor of automorphy on an abelian variety to a divisor

Given a complex abelian variety $A = V/\Gamma$ (for $\Gamma$ being a lattice in the complex vector space $V$), one knows how to describe a holomorphic line bundle in terms of factors of automorphy: By ...

**4**

votes

**0**answers

122 views

### Computing theta functions of lattices in practice

I am motivated by a problem in 2d CFT to compute "generalized theta functions," expressions of the form
\begin{equation}
\vartheta_{L,u}(\tau) := \sum_{\alpha \in L} u(\alpha) q^{{\langle\...

**0**

votes

**0**answers

72 views

### The loss of double periodicity (ellipticity)

Consider a meromorphic function $f(\mathfrak{a}_1, \mathfrak{a}_2)$, such that
$$
\begin{align}
f(\mathfrak{a}_1, \mathfrak{a}_2) = f (\mathfrak{a}_1 + 1, \mathfrak{a}_2) = f(\mathfrak{a}_1 + \tau, \...

**0**

votes

**1**answer

109 views

### Obtain a series expansion of $a^2(q)a^2(q^4)$

Let $a(q)$ denote the Borwein function $$a(q)=\sum_{m,n=-\infty}^\infty q^{n^2+nm+m^2}.$$
In this research paper the author has obtained the series expansion for $a(q)a(q^4)$. I want the series ...

**5**

votes

**0**answers

201 views

### 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+...

**10**

votes

**0**answers

290 views

### When exactly is the principal AGM equal to the optimal AGM?

Definitions
Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...

**12**

votes

**0**answers

275 views

### Work of Atkin on the 26th power of eta

The 26th power of the Dedekind $\eta$ function has been mentioned several times here on MO:
A 14th and 26th-power Dedekind eta function identity?
What's the status of the following relationship ...

**3**

votes

**1**answer

162 views

### Can the following sum be counted or expressed in terms of special functions?

Let us define this sum as a function of $z \in \mathbb{C}$ with some positive parameter $a$
$$
f(z; a) = \sum\limits_{n = 0}^{\infty}\frac{|z|^{2n}}{n!}e^{-ian^2}.
$$
Probably, it can be expressed (or ...

**5**

votes

**1**answer

157 views

### Jacobi forms and Kato's modular units

this is pretty much just a silly literature question; apologies in advance. Kato uses the following theta function (or slight variants thereof) in his construction of his Euler system:
$$\Theta(\tau, ...

**0**

votes

**0**answers

61 views

### Proving the Immersion part of an Embedding

Trying to see the proof of embedding the Jacobian of a Compact Riemann Surface $X$ using Theta functions. So, using the Theta divisor we have the corresponding line bundle say $L$, we want to prove ...

**2**

votes

**1**answer

140 views

### Semidirect product of metaplectic group and Heisenberg group

I know that Symplectic group has an action on Heisenberg group.
I am wondering how to extend this to non-trivial two fold metaplectic covering?
Thanks in advance!

**1**

vote

**0**answers

67 views

### Construct a doubly periodic function on $\mathbb{C}$ using Jacobi elliptic functions with anti-holomorphic involution

I would like to find explicit examples of non-constant meromorphic doubly periodic $f(z)$ on $\mathbb{C}$, i.e. a meromorphic function on $\mathbb{C} / \Gamma$ for some lattice $\Gamma$, such that ...

**3**

votes

**1**answer

91 views

### " Laurent expansion" of quasi-periodic complex complex function

Suppose a complex function $f(z)$ depends only on $z$, and satisfies the quasi-periodicity in both directions:
$$f(z+ a_x)= e^{i \theta_{a_x}} f(z)$$
$$f(z+ i a_y)= e^{i \theta_{a_y}} f(z)$$
where $\...

**10**

votes

**1**answer

2k views

### Reference request: proof of Ramanujan's Cos/Cosh Identity

The Ramanujan Cos/Cosh Identity, as stated here, is
$$\left[1+2\sum_{n=1}^{\infty}\frac{\cos n\theta}{\cosh n\pi}\right]^{-2}+
\left[1+2\sum_{n=1}^{\infty}\frac{\cosh n\theta}{\cosh n\pi}\right]^{-2}=...

**16**

votes

**0**answers

477 views

### Finite version special case Jacobi triple product formula

In this paper, Shanks uses the following formula:
$$ \sum_{s=0}^{n-1}q^{s(2n+1)} \times \left( \prod_{k=s+1}^{n} \dfrac{1-q^{2k}}{1-q^{2k-1}}\right) = \sum_{s=1}^{2n} q^{\frac{s(s-1)}{2}}$$
to get a ...

**2**

votes

**0**answers

146 views

### Rationality of Eisenstein series g2 and g3 for elliptic curves defined over numberfields

Let $K$ be a number field and let $E/K$ be an elliptic curve. (Fix an embedding of $K$ into the complex numbers $\mathbb{C}$). Let $\eta$ be the invariant differential of $E/K$. Let $\omega_1$ and $\...

**1**

vote

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71 views

### Decomposing functions on the fundamental domain of the torus into cusp forms, eisenstein series

I am trying to do some elementary calculations to understand the properties of the following spectral resolution on $H/SL(2,\mathbf{Z})$. (Half plane mod modular group; fundamental domain of the torus)...

**3**

votes

**0**answers

95 views

### An elliptic function built from a log-theta-function integral?

I'm studying the apparent ellipticity of $$\Theta(z,a):=\frac{\exp(\tfrac2a\int_0^a\ln\vartheta(x+z)\,dx)}{\vartheta(z)\vartheta(z+a)},\tag1$$with $a$ is a free parameter and \begin{align}\vartheta(z)&...

**2**

votes

**0**answers

108 views

### theta function with a low bound in the sum

I encounter a sum similar to the Jacobi Theta function except there is a lower bound $-J$ with $J\geq 0$:
\begin{equation}
f(q,x)=\sum_{n=-J}^\infty q^{n^2}x^n.
\end{equation}
My question is whether ...

**5**

votes

**1**answer

206 views

### "One half of a theta-function" - is there something in the literature about it?

In an answer to my question Enumeration of lattice paths of a specific type (which was in turn caused by an older one, "Special" meanders) I encountered the series
$$
F(t,q):=\sum_{n=1}^\...

**7**

votes

**0**answers

129 views

### Random walk on $\mathbf{Z}_d$ with Jacobi $\theta$ transition probabilities

In the context of a finite-dimensional quantum mechanical problem, I was led to study the random walk on $\mathbf{Z}_d$ (i.e the integers modulo $d$), $d$ odd with transition probabilities given by:
$...

**2**

votes

**0**answers

104 views

### Algebraic theta-functions of level $2$ on an elliptic curve

Consider an elliptic curve $E\!: y^2 = x(x-1)(x-\lambda)$, where $\lambda \in k \setminus \{0, 1\}$ for some field $k$ of $\mathrm{char}(k) \neq 2$. Also consider the ample divisor $D = 2P_0$, where $...

**2**

votes

**0**answers

99 views

### Understanding proof of q-theta function expression

In arXiv:math/0309252v4 at the bottom of page 11, the following result is proposed
$$ b_1 \theta(b_0 b_1^{\pm};p) \frac{z^{-1}\theta(z^2;p)}{\theta(b_0 z^{\pm};p)\theta(b_1 z^{\pm};p)} = (p;p)^{-2}\...

**3**

votes

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93 views

### Evaluating a Fermi gas problem for a SO(2N+1) matrix integral

I have the following multiple integral derived from a random matrix calculation I wish to evaluate
$$\int_0^{\pi} dx_1 dx_2 \cdots dx_n \rho(x_1,x_2)\cdots \rho(x_n,x_1)$$
where the $\rho$ functions ...

**4**

votes

**1**answer

256 views

### Weight, Index, and Congruence Subgroup of Classical Jacobi Theta Functions

On the very first page in the Introduction of Eichler and Zagier's text on Jacobi forms, they mention that the theta function
$$\Theta_{x_{0}}(\tau, z) = \sum_{x \in \mathbb{Z}^{N}} q^{Q(x)} y^{B(x, ...

**0**

votes

**0**answers

74 views

### Deriving "quasi-theta" functions from theta functions' zeros

I've been trying to sift the zeros of \begin{align}\vartheta(z)&=e^{-\pi z^2}\prod_{k\ge1}(1+e^{-(2k-1)\pi+2\pi z})(1+e^{-(2k-1)\pi-2\pi z})\\
&=\prod_{k\ge1}(1+e^{-(2k-1)\pi+2i\pi z})(1+e^{-(...

**5**

votes

**1**answer

333 views

### $q$-analog of an integral from quantum field theory?

This question has been completely reformulated and a new property for the function $f_q$ has been added due to a series of helpful comments by fedja.
Consider the integral from quantum field theory ...

**1**

vote

**1**answer

192 views

### How does $\prod_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^5) $ hold?

While I check the proof for entry 10(vii) in Chapter 19 in Ramanujan's notebook, I couldn't understand one equality. It is
\begin{equation}
\prod_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^...

**2**

votes

**1**answer

162 views

### "Sparse" Theta Series

The number of integer points with a given norm in the integer grid $\mathbb{Z} \times \mathbb{Z}$ may be calculated via the generating function
$$\theta_3(q)^2= \left(\sum_{n \in \mathbb{Z}} q^{n^2}\...

**4**

votes

**0**answers

172 views

### A statement in Siegel's paper on Riemann-Siegel Formula

Siegel left the following comment in the last paragraph of his famous "Über Riemanns Nachlaß zur analytischen Zahlentheorie", Section 1:
Das Integral $\Phi(u)$ ist ein Spezialfall des Integrales $$\...

**1**

vote

**0**answers

122 views

### Basic properties of prime forms

The prime form of a Riemann surface of genus $g$ can be defined in terms of multi-dimensional $\vartheta$-functions as follows:
$$E(z,w) = \frac{\vartheta\Big[\genfrac{}{}{0pt}{}{\vec{a}}{\vec{b}} \...

**4**

votes

**0**answers

278 views

### modularity of Theta functions attached to Hecke characters

Let $K/\mathbb{Q}$ be a quadratic imaginary field, and let $\chi$ be a Hecke character on $K$. Using Poisson summation, one can show that the theta function
$$
\theta(z):=\sum_{I\subseteq \mathcal{O}...

**6**

votes

**3**answers

546 views

### Alternating power series $\sum_{k\geq 0}(-1)^k z^{(2k+1)^2}$

Suppose that $f(x):=\sum_{k\geq 0}(-1)^ke^{-(2k+1)^2x}$ has a holomorphic continuation to a neighborhood of $0$, that is, $f(x)=\sum_{n\geq 0}a_n x^n$ for $x> 0$ small. I want to know the value of $...

**1**

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**0**answers

145 views

### Theta functions, a natural basis.

Let $L$ be the principal polarization of an abelian variety $A$. Some people say that the theta functions with characteristics form a natural basis of the global section $\Gamma(A,L^{\otimes n})$ for $...

**4**

votes

**0**answers

212 views

### Expansion of Jacobi theta function at $p$-torsion

I am aware of the formula $$\Theta(z,q)=z\exp\left( -2\sum_{k\geq 1} \frac{z^{2k}}{(2k)!}E_{2k}(q)\right)$$ for the Jacobi theta function at the origin $z=0$. The definition I am using for the theta ...

**17**

votes

**1**answer

946 views

### Theta functions, re-expressed

Recall the classical $\theta(q):=\prod_{k=1}^{\infty}(1-q^k)$ and
define the sequences $a_n$ and $b_n$ by
$$\frac{\theta^3(q)}{\theta(q^3)}=\sum_{n=0}^{\infty}a_nq^n \qquad \text{and} \qquad
F(q):=\...

**3**

votes

**1**answer

569 views

### On $x^k+y^k=1$ and the Dixonian elliptic functions

This solves this post and is also related to this MO post by involving $\tfrac12,\tfrac13,\tfrac14,\tfrac16$.
$p=2$
The singly-periodic trigonometric functions and the doubly-periodic Jacobi ...

**3**

votes

**2**answers

517 views

### Inverse Laplace transform of $sech(\sqrt{2\lambda})$ and Brownian motion occupation time

In dimension 3 we have that for $T=\int_{[0,\infty)}1_{B_{t}\in B(0,1)}dt$ has the Laplace transform
$$E[e^{-\lambda T}]=sech(\sqrt{2\lambda}).$$
And in dimension 1 we have the same for $\tau=\min\{t:...

**6**

votes

**1**answer

306 views

### Finding order of vanishing for Jacobi Theta function

From Rademacher's book (Topics in Analytic Number Theory) I'm using the functional equation of $\vartheta_2(0|\tau) = 2\sum_{m=0}^\infty q^{\left(m+\frac12\right)^2} = \vartheta_2(\tau)$ and the fact ...

**13**

votes

**2**answers

668 views

### A conjecture about algebraic values of $(-q;\,-q)_\infty/(q;\,q)_\infty$

Recall that $(a;\,q)_\infty$ is the $q$-Pochhammer symbol:
$$(a;\,q)_\infty=\prod_{n=0}^\infty(1-a \, q^n).\tag1$$
Its important special case $(q;\,q)_\infty=\prod_{n=1}^\infty(1-q^n)$ is sometimes ...

**4**

votes

**0**answers

96 views

### Triple Petersson Inner Products With Theta Functions

Our current work requires us to bound triple products of the form
$$\langle \Im(*)^{\frac{3}{2}}|\theta^3|^2,\mu_j \rangle,$$
where $\langle \cdot, \cdot \rangle$
is the Petersson inner product, ...

**7**

votes

**1**answer

309 views

### Modularity of certain theta series associated to hyperbolic lattice

Let $L$ be an even hyperbolic lattice, i.e. a free $\mathbb{Z}$-module with a non-degenerate inner product $\cdot$ valued in $\mathbb{Z}$ of signature $(1,n)$ such that the norm of every vector is ...

**4**

votes

**1**answer

128 views

### Modular S transformation on higher order $\vartheta$-functions using Poisson summation

I am trying to understand the action of a modular S transformation on a $\vartheta$-function. To do this for the problem I'm considering I first need to understand the following.
Given a $\vartheta$-...

**1**

vote

**0**answers

82 views

### Developing a functional equation for log-integral of theta function

I'm studying $u(z,q):=\exp(\left[\tfrac{\pi q}{12}-\ln2-\sum_{k\ge1}\ln(1-e^{-2k\pi q})\right]z+\int_1^z\ln\varphi(x,q)\,dx)$ , with $$\varphi(z,q)=2e^{-\pi qz^2-\pi q/4}\sinh \pi qz\prod_{k\ge1}(1-e^{...

**2**

votes

**2**answers

489 views

### Lower bound for the number of representations of integers as sum of squares

Let $k\geq 4$. As usual, let $r_k(n)$ denote the number of ways to represent $n$ as the sum of $k$ squares. Is this true that for every $\varepsilon>0$, one has $r_k(n) \gg n^{\frac{k}{2}-1-\...

**7**

votes

**2**answers

438 views

### How to estimate a specific infinite matrix 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 $k$ ...

**3**

votes

**0**answers

80 views

### Log-concavity of difference of theta functions

My knowledge on theta functions is limited, but I suspect that this is a quite challenging question. The 3rd Jacobian Theta function is given by
\begin{equation}
\theta_3(z,q)\,=\,\sum\limits_{n=-\...

**3**

votes

**0**answers

167 views

### Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...

**5**

votes

**1**answer

454 views

### Summation of an infinite q-series

When calculating a Partition function, I encounter the following summation
$$\sum_{n=0}^{\infty} x^n q^{n^2}.$$
I know that the sum $\sum_{n=-\infty}^{\infty} x^n q^{n^2}$ is a Theta function, but I ...