Questions tagged [theta-functions]

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

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Residue theorem and theta function identities

Let's use the classical definition $$ \vartheta _1\left( z,q \right) =-i\sum_{n\in \mathbb{Z}}{\left( -1 \right) ^nq^{\left( n+\frac{1}{2} \right) ^2}e^{i\left( 2n+1 \right) z}}\,\,\,\,\,\ q=e^{i\pi \...
Loyar's user avatar
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How to prove the equality of power series below?

Assume $$ F\left( x \right) := \sum_{n=-\infty}^{+\infty}{x^{\left( n+\frac{1}{2} \right) ^2}}, G\left( x \right) := \sum_{n=-\infty}^{+\infty}{x^{n^2}}, H\left( x \right) := \sum_{n=-\infty}^{+\infty}...
SHBooKP's user avatar
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What is the transformation formula of Theta series associated the quadratic form over totally real field?

I am currently reading the book of Erich Hecke "Lectures on the Theory of Algebraic Numbers", and it is not clear for me to understand his notation of theta series associated to quadratic ...
Vector's user avatar
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Mistake computing $\sum_{n=1}^{+\infty} \frac{n}{e^{2\pi n}-1} = \frac{1}{24}-\frac{1}{8\pi}$

I recently gave a try to show that $$\sum_{n=1}^{+\infty} \frac{n}{e^{2\pi n}-1}=\frac{1}{24}-\frac{1}{8\pi} $$ without using the Theta function or Mellin transform, but I ended up with twice the ...
azur's user avatar
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Understanding the proof of Theorem 10.1 in Montgomery & Vaughan's Multiplicative Number Theory

In the last step of the proof of Theorem 10.1 in the book Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan I couldn't understand what exactly "turn the ...
Ali's user avatar
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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
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Close form expression for an integral with z derivative of jacobi theta function

I have an expression of the form $$ \tag{1} A(\chi) = \int_{0}^\infty\sum_{i=0}^\infty (-1)^{i+1}\frac{(2i+1)\chi}{\sqrt{t}}\exp\left(\frac{-(2i+1)^2\chi^2}{t}\right)\mathrm{d}t. $$ If I am not ...
ck1987pd's user avatar
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Does the theory of theta functions ever go beyond just random caulculations and ugly formulas?

I recently took a class in elliptic functions and found the theory of elliptic functions very clean. You get very powerful results and the proofs are usually short and intuitive. Then we learned about ...
akin's user avatar
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How to compute $\sum_{n=0}^{\infty} \frac{(-1)^n}{\cosh((n+\frac{1}{2}) \pi)}$

How to compute $\sum_{n=0}^{\infty} \frac{(-1)^n}{\cosh((n+\frac{1}{2}) \pi)}$? My attempt was trying to consider the Mellin transform of $\frac{1}{\cosh(x)}$ and use the inverse Mellin transform as a ...
Dqrksun's user avatar
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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
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An explicit basis of space of modular forms $M_5(\Gamma_0(11),(\frac{\bullet}{11}))$

I want to find a basis of space of modular forms in $M_5\left(\Gamma_0(11),\left(\frac{\bullet}{11}\right)\right)$, which is of dimension $5$. Additionally, I want the basis to have explicit forms. ...
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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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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
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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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Product of Jacobi theta functions

I need to calculate the following quantity: $$ \zeta(x;\mu_1,\mu_2,\tau) = πœ—_3({x-\mu_1},𝜏) \cdot πœ—_3({x-\mu_2},𝜏) $$ $πœ—_3(𝑧,𝜏)$ being the Jacobi theta function defined as: $$ πœ—_3(𝑧,𝜏)=βˆ‘_{𝑛=...
AleMarco's user avatar
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Ramanujan's identity concerning a quotient of Dedekind's eta functions

In his paper On certain Arithmetical Functions (published in Transactions of the Cambridge Philosophical Society, XXII, No. 9, 1916, pp. 159-184) Ramanujan presents the following identities (as if ...
Paramanand Singh's user avatar
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An identity related to $q$-series

While studying Ramanujan's theta functions, I encountered a q-series $(q;q)_\infty^2\phi(q)$. I calculated the first few terms of $(q;q)_\infty^2\phi(q)$ and observed that it seems to have the ...
Kevin's user avatar
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Proving a Functional Equation for the JacobiTheta Function

Let $\Theta(t) = \sum_{k = - \infty}^{\infty} e^{-\pi k^{2} t} $. How can it be proved that $\Theta(\frac{1}{t}) = \sqrt{t}\Theta(t)$? I have read a proof here https://scholarship.claremont.edu/cgi/...
Robert's user avatar
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$\Theta$-notation of the recurrences $T(n) = 3T(n/2 +1) + n$ and $T(n) = 4T(n/2)-4T(n/4)+1$ [closed]

(a) $T(n) = 3T \left ( \frac{n}{2} + 1 \right ) + n$ (b) $T(n) = 4T \left ( \frac{n}{2} \right ) - 4 T \left ( \frac{n}{4} \right ) + 1$ I am really stuck on these two recurrences and finding out ...
user123's user avatar
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Approximations with Theta and Dawson Functions

Consider the following function \begin{align} g(x)=e^{-x/4}\int_0^{x/4}e^y\vartheta(2\textstyle\sqrt{y/\pi})dy, \end{align} where $\vartheta$ is the theta function, given by $$ \vartheta(x)=\sum_{n\in\...
sam wolfe's user avatar
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9 votes
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Ratio of theta functions as roots of polynomials

I was playing with the theta functions with argument $ z = 0 $ $ \vartheta_2(q) =\sum_{n=-\infty}^\infty q^{(n+1/2)^2} $ $ \vartheta_3(q) =\sum_{n=-\infty}^\infty q^{n^2} $ $ \vartheta_4(q) =\sum_{n=-\...
user967210's user avatar
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Ratio of theta function derivatives with theta function

I have the following ratios I want to compute. $$ \frac{ \left( \frac{\partial \vartheta_3(v, q)}{\partial v} \right)^2 }{C + \left(\vartheta_3(v, q)\right)^2 }, $$ where $C$ is a constant. $$ \frac{ \...
CfourPiO's user avatar
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Surjectivity of Weil representation

Let $F$ be a local field (e.g. $\mathbb{R}$, $\mathbb{C}$, or finite extensions of $\mathbb{Q}_p$). Let $X$ (resp. $Y$) be a non-degenerate quadratic (resp. symplectic) space over $F$. Then $\mathrm{...
Seewoo Lee's user avatar
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Proving the existence of a summation and the empirical results of it

I have a summation of the form: $$ S = \sum_{i=1}^{\infty} \sqrt{\frac{2}{\pi i}} \vartheta_4\left(0,\exp\left(\frac{-2H^2}{i}\right)\right), $$ where $\vartheta_4(z=0,q)$ is the fourth jacobi theta ...
ck1987pd's user avatar
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Theta Functions in $\mathbb{C} \cdot \Delta$ and Spanning $M_{4k}(\mathrm{SL}{2}(\mathbb{Z}), \vartheta{\text {tr }})$

I am working on a problem related to quadratic forms and theta functions. I have the following two part question: a) Show that there are $A_{1}, A_{2} \in SP_{24}$ with $\operatorname{det}\left(A_{1}\...
Mathematiker's user avatar
3 votes
1 answer
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Exercise in the book "Basic Hypergeometric Series" of Gasper and Rahman

I am trying to solve Exercise 1.12.(iii) from the book "Basic Hypergeometric Series" of Gasper and Rahman (see the picture below). I am especially interested in the case where $c=ab$ and $n=...
Stabilo's user avatar
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6 votes
3 answers
307 views

Closed form for $\sum_{n=1}^\infty \frac{1}{n(e^{2 n \pi}-1)}$

I need to calculate the sum $$\sum_{n=1}^\infty \frac{1}{n(e^{2 n \pi}-1)}$$ Write $$S=\sum_{n=1}^\infty \frac{1}{n(e^{2 n \pi}-1)}$$ $$S=\sum_{n=1}^\infty\frac{1}{n} \sum_{m=1}^\infty e^{-2 \pi n m}$$...
Max's user avatar
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3 votes
0 answers
132 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
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1 answer
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Basic properties of Jacobi's theta function

On the following Wikipedia page about Jacobi's theta function $\vartheta$, it says that the $\vartheta$ satisfies the condition that "at fixed $\tau$, this is a Fourier series for a $1$-periodic ...
Squirrel-Power's user avatar
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Additional symmetries in a theta-like function

Let $\Theta : \mathfrak{h}\times \mathfrak{h} \to \mathbb{R}$ be defined as follows $$ \Theta(z,\tau) = \sum_{\omega_1, \omega_2 \ \in \ \mathbb{Z}\tau + \mathbb{Z}} \exp\Big(-2\pi\frac{| \...
Testcase's user avatar
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1 answer
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Alternative expression for theta function

I've been trying to re-express the particular theta function $$\theta(r)=\frac{1}{2}\sum_{n\in\mathbb{Z}}\cos(nr)\exp(-n^2/\lambda)$$ in the form $$\theta(r)=\sum_{n\in\mathbb{Z}}\alpha_n\exp(-\lambda(...
arnold's user avatar
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6 votes
2 answers
307 views

Conjectured identity for the ratio of Ramanujan theta functions

Following Ramanujan, we define theta functions as follows $$\chi(q):=\prod_{n = 1}^{\infty}\left(1+q^{2n-1}\right),\\\phi(q)=\sum_{n=-\infty}^{\infty}q^{n^2},\\\displaystyle \psi(q)=\sum_{n = 0}^{\...
Nicco's user avatar
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6 votes
1 answer
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Integrals of Jacobi $\vartheta$ functions on the interval $[1,+\infty)$

I start from the following obvious observation, which is declared to be($q=e^{-\pi x}$): \begin{aligned} \int_{1}^{\infty}x\vartheta_2(q)^4\vartheta_4(q)^4 \text{d}x&=\int_{0}^{1}x\vartheta_2(q)^4\...
Setness Ramesory's user avatar
2 votes
0 answers
80 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
6 votes
4 answers
238 views

A theta function identity involving $\vartheta_2(q^3),\vartheta_3(q^3)$

How can we verify the theta function identity? $$ \left ( \vartheta_2(q)^2+3\vartheta_2(q^3) ^2\right )\left ( \vartheta_3(q)^2+3\vartheta_3(q^3)^2 \right )=4\vartheta_2(q)^2\vartheta_3(q)^2. $$ Where ...
Setness Ramesory's user avatar
0 votes
1 answer
44 views

Showing an equality for the KS-statistic with Jacobi theta functions

I recently read about the Kolmogorov-Smirnov statistic and on wikipedia it is stated that the cdf of the random variable $K$ is given by $$ Pr(K\leq x) = 1-2\sum_{n=1}^{\infty} (-1)^{n-1}e^{-2n^2x^2} =...
AxelT's user avatar
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What is the exact value of $\sum\limits_{n=1}^\infty \frac{1}{F_n}$? [duplicate]

What is the exact value of $\sum\limits_{n=1}^\infty \frac{1}{F_n}$? $F_n$ denotes the $n^{th}$ Fibonacci number. Wolframalpha gave me this answer: $$\sum_{n=1}^{\infty}\frac{1}{F_n}\ =\frac{1}{4}\...
Dylan Levine's user avatar
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Do closed form expressions exist for $\int_0^1 \theta_3(x)\,dx$?

The Jacobi theta (or β€œthetanull”) function $\theta_3$ is defined by: $$\theta_3(x)= \sum_{n \in \mathbb{Z}} \mathrm{e}^{-\pi n^2 x} = 1+ 2\sum_{n \in \mathbb{N}} \mathrm{e}^{-\pi n^2 x} \qquad \Re(x) &...
Agno's user avatar
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1 vote
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Checking the parity of an infinite sum

While going through a paper I stumbled on the following equality : $\begin{equation} \rho(\alpha)=\frac{1}{\pi}\sum\limits_{k\in\mathbb{Z}}\frac{e^{2ik\alpha}}{\cosh(k\zeta)}=\frac{1}{\pi}\frac{\...
Ceethemez's user avatar
9 votes
2 answers
370 views

Prove a relation between Theta functions and their derivatives to prove $\sum_{n=-\infty}^\infty \frac{1}{\cosh^2(\pi n )}$

In an attempt of proving \begin{align*} \sum_{n=-\infty}^\infty \frac{1}{\cosh^2(\pi n )}=\frac{1}{\pi}+\frac{\Gamma^4\left( \frac14\right)}{8 \pi^3} \tag{1} \end{align*} I found the following ...
Ricardo770's user avatar
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7 votes
0 answers
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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
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62 views

Why does the theta function appear in the study of complex tori?

Let $L$ be a lattice and $X = \mathbb{C}/L$ a complex tori. To obtain a meromorphic function $f$ on $X$, it is standard to consider ratios of products of (translated) theta-functions, and in fact, any ...
ferhenk's user avatar
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$(\mathfrak{g}, K)$-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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2 votes
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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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8 votes
1 answer
242 views

Mellin transform of theta function $\theta$ to show $\zeta(-1)=-\frac{1}{12}$

Edit: I realize a lot of attention is pointed at the proclamation $\zeta(-1)=-\frac{1}{12}$. Im more interested in the derivation of the Zeta function using the methods describe below, thank you! ...
Batmannilsson's user avatar
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0 answers
66 views

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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3 votes
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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
4 votes
1 answer
353 views

How to prove Gauss's identities on the action of the operator $f' = x\frac{df}{dx}$ on Jacobi theta functions?

P.444-445 of volume 3 of Gauss's Nachlass, in which Gauss wrote down an identity for the infinite series of $\vartheta^4_3(x)$ (this identity is the essence of Jacobi's four squares theorem), include ...
user2554's user avatar
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4 votes
3 answers
338 views

How to prove an identity of Gauss for the logarithm of Jacobi theta function?

P.444 of volume 3 of Gauss's Nachlass includes the following identity for the logarithm of $\vartheta_3(x)$: $$\mathbb{log}(1+2x+2x^4+2x^9+\cdots) = \frac{2x}{1+x}+\frac{2x^3}{3(1+x^3)}+\frac{2x^5}{5(...
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