# Questions tagged [theta-functions]

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

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• 194
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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 ...
• 275
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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 ...
• 167
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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 ...
• 183
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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 ...
1 vote
138 views

### 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 ...
• 1,104
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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 ...
• 97
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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 ...
• 442
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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 ...
• 3,230
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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. ...
• 879
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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 ...
• 717
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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 ...
• 717
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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$E=\mathbb{C}/(\tau\mathbb{Z}+\mathbb{Z})\to \mathbb{P}^2$ ...
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1 vote
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• 3,427
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### 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 ...
• 2,761
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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{...
• 878
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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 ...
• 54.6k
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### 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 ...
• 457
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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 ...
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• 1,407
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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 ...
• 846
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### 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 ...
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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(...