# Tagged Questions

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 ...
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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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### 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 ...
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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 ...
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### conjectured identity of the product of two theta functions

Looking into the discussion in this post,I was naturally led to consider the following general identity Given the two jacobi theta functions,$$\theta_2(q)=\sum_{n=-\infty}^\infty q^{(n+1/2)^2}$$ and ...
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### 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 ...
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### Primitive of Weierstrass $\wp$

Consider a lattice $L=\mathbb{Z}+\mathbb{Z}\tau$. Take the function $\xi(z) = \frac{-1}{z} - \sum_{w \in L\backslash \{0\}} \Big ( \frac{1}{z-w} + \frac{1}{w} + \frac{z}{w^2} \Big )$. Obviously this ...
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### 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 ...
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### Zeros of the derivative of a theta function

I'm trying to find the positions or at least the number of zeros of $\partial_z\vartheta[p,0](z,\tau)$ where the standard theta function with characteristics $(p,q)$ is defined as ...
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### Jacobi Theta Functions?

For the Jacobi theta function $\vartheta_3(z|\tau)$ there exists an equality (by Whittaker & Watson) \vartheta_3(z|\tau) = \sum_{n=-\infty}^{\infty} e^{n^2 \pi i \tau + 2 n i z} ...
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### Motivation behind theta function calculation in Diamond & Shurman

I am reading section 1.2 of A First Course in Modular Forms. Let $q=e^{2\pi i\tau}$, where $\tau\in\cal H$ is assumed to be in the upper half plane, and define $\theta(\tau)=\sum_{n\in\Bbb Z}q^{n^2}$. ...
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### Theta-divisor and special divisors

I'm studying the Jacobi inversion problem. Could you help me with the next question? Let us consider an algebraic curve $C$ and it's Jacobian $J(C)$. For a initial point $P_0$ we can construct the ...
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### What is meant by a set of generic points on a compact Riemann surface

Let $X$ be a compact connected Riemann surface of genus $g \geq 1$. I'm studying a theorem of Faltings which looks as follows. Let $P_1,\ldots, P_g$ be generic points on $X$. Then we have some ...
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