The <Jacobi triple product> gives
$$
\theta(\tau)=\prod_{n\geq1}(1-q^{2n})(1+q^{2n-1})^2,
\qquad q=e^{\pi i\tau}.
$$
Every factor is nonzero for $|q|<1$, and the product converges to a nonzero limit, so $\theta$ has no zero in the upper half-plane.

Nonvanishing of the Jacobi theta function

The specialization of the <Jacobi triple product> to the <Jacobi theta function> is
$$
\theta(\tau)=\prod_{n\geq1}(1-q^{2n})(1+q^{2n-1})^2,
\qquad q=e^{\pi i\tau}.
$$
Since $\tau\in\mathfrak h$, one has $|q|<1$, so every displayed factor is nonzero. Moreover
$$
\sum_{n\geq1}\bigl(|q|^{2n}+2|q|^{2n-1}\bigr)<\infty.
$$
The standard convergence criterion for <infinite products> therefore shows that the product converges to a nonzero value. This proves the <nonvanishing of the Jacobi theta function> throughout $\mathfrak h$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 3 / d / Solution

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For $|q|<1$ and $z\ne0$, the Jacobi triple product is
$$
\sum_{n\in\mathbb Z}z^nq^{n^2}
=\prod_{m\geq1}(1-q^{2m})(1+zq^{2m-1})(1+z^{-1}q^{2m-1}).
$$

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