# Questions tagged [theta-functions]

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### 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 ...
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### 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 ...
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### 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\...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
### 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 $\vartheta$-...