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Problem $1$. Which full rank lattices $\Lambda \subset \mathbb R^d$ have their corresponding theta function $\theta_{\Lambda}(\tau):= \sum_{\bf n \in \Lambda } e^{\pi i \tau ||n||^2}$ zero-free in the complex upper half plane $\mathbb H$?

For example, for $d=1$, the classical theta function $\theta_{\mathbb Z}(\tau):= \sum_{\bf n \in \mathbb Z } e^{\pi i \tau n^2}$ is the ratio of two Dedekind eta functions, so it is the ratio of two infinite products in q, and hence zero-free inside $\mathbb H$. Similarly, any theta function that can be written as an infinite product in a "nice way" is zero-free.

What I would really like is a criterion that, instead of brute-forcing a contour integral to search for zeros, gives us some nice conditions on the lattice $\Lambda$, involving such parameters such as, for example, $vol( \Lambda)$, the successive minima of $\Lambda$ relative to the unit ball, the dual lattice, $Aut(\Lambda)$, etc.

EDIT (on Nov 16, 2012). To consider more examples, fix any dimension $d$ and consider any diagonal positive definite quadratic form $a_1 x_1^2 + \dots + a_d x_d^2$, with $a_j$ any fixed positive real numbers. The corresponding lattice $\Lambda$ is therefore the direct sum of $d$ one-dimensional lattices, namely $\sqrt a_1 \mathbb Z \oplus \cdots \oplus \sqrt a_d \mathbb Z$, so that $\theta_\Lambda$ is the product of $d$ one-dimensional theta functions, and hence $\theta_\Lambda$ is nonzero in $\mathbb H$.

More generally, if $\Lambda$ is reducible in the sense that it is expressible as the direct sum of some lower dim'l lattices, then we call the corresponding theta function reducible. It is therefore natural to ask the following more particular question.

Problem 2. Which irreducible theta functions $\theta_{\Lambda}(\tau)$ are zero-free in $\mathbb H$?

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Problem $1$. Which full rank lattices $\Lambda \subset \mathbb R^d$ have their corresponding theta function $\theta_{\Lambda}(\tau):= \sum_{\bf n \in \Lambda } e^{\pi i \tau ||n||^2}$ zero-free in the complex upper half plane $\mathbb H$?

For example, for $d=1$, the classical theta function $\theta_{\mathbb Z}(\tau):= \sum_{\bf n \in \mathbb Z } e^{\pi i \tau n^2}$ is the ratio of two Dedekind eta functions, so it is the ratio of two infinite products in q, and hence zero-free inside $\mathbb H$. Similarly, any theta function that can be written as an infinite product in a "nice way" is zero-free.

What I would really like is a criterion that, instead of brute-forcing a contour integral to search for zeros, gives us some nice conditions on the lattice $\Lambda$, involving such parameters such as, for example, $vol( \Lambda)$, the successive minima of $\Lambda$ relative to the unit ball, the dual lattice, $Aut(\Lambda)$, etc.

EDIT (on Nov 16, 2012). To consider more examples, fix any dimension $d$ and consider any diagonal positive definite quadratic form $a_1 x_1^2 + \dots + a_d x_d^2$, with $a_j$ any fixed integers (thus allowing both definite and indefinite forms)positive real numbers. The corresponding lattice $\Lambda$ is therefore the direct sum of $d$ one-dimensional lattices, namely $a_1 \mathbb Z \oplus \cdots \oplus a_d \mathbb Z$, so that $\theta_\Lambda$ is the product of $d$ one-dimensional theta functions, and hence $\theta_\Lambda$ is nonzero in $\mathbb H$.

More generally, if $\Lambda$ is reducible in the sense that it is expressible as the direct sum of some lower dim'l lattices, then we call the corresponding theta function reducible. It is therefore natural to ask the following more particular question.

Problem 2. Which irreducible theta functions $\theta_{\Lambda}(\tau)$ are zero-free in $\mathbb H$?

Problem $1$. Which full rank lattices $\Lambda \subset \mathbb R^d$ have their corresponding theta function $\theta_{\Lambda}(\tau):= \sum_{\bf n \in \Lambda } e^{\pi i \tau ||n||^2}$ zero-free in the complex upper half plane $\mathbb H$?

For example, for $d=1$, the classical theta function $\theta_{\mathbb Z}(\tau):= \sum_{\bf n \in \mathbb Z } e^{\pi i \tau n^2}$ is the ratio of two Dedekind eta functions, so it is the ratio of two infinite products in q, and hence zero-free inside $\mathbb H$. Similarly, any theta function that can be written as an infinite product in a "nice way" is zero-free.

What I would really like is a criterion that, instead of brute-forcing a contour integral to search for zeros, gives us some nice conditions on the lattice $\Lambda$, involving such parameters such as, for example, $vol( \Lambda)$, the successive minima of $\Lambda$ relative to the unit ball, the dual lattice, $Aut(\Lambda)$, etc.

EDIT (on Nov 16, 2012). To consider more examples, fix any dimension $d$ and consider any diagonal quadratic form $a_1 x_1^2 + \dots + a_d x_d^2$, with $a_j$ any fixed integers (thus allowing both definite and indefinite forms). The corresponding lattice $\Lambda$ is therefore the direct sum of $d$ one-dimensional lattices, namely $a_1 \mathbb Z \oplus \cdots \oplus a_d \mathbb Z$, so that $\theta_\Lambda$ is the product of $d$ one-dimensional theta functions, and hence $\theta_\Lambda$ is nonzero in $\mathbb H$.

More generally, if $\Lambda$ is reducible in the sense that it is expressible as the direct sum of some lower dim'l lattices, then we call the corresponding theta function reducible. It is therefore natural to ask the following more particular question.

Problem 2. Which irreducible theta functions $\theta_{\Lambda}(\tau)$ are zero-free in $\mathbb H$?