= Period lattice of theta3 over theta4
{title2=$\operatorname{Per}(\theta_3/\theta_4)=\pi\mathbb Z+2\pi\tau\mathbb Z$}
Use the radian spatial convention $\theta_3(z,\tau)=\sum_n e^{\pi in^2\tau+2inz}$ and $\theta_4(z,\tau)=\theta_3(z+\pi/2,\tau)$. Their ratio is unchanged by $\pi$ and changes sign under $\pi\tau$. The numerator has simple zeros at $\pi(1+\tau)/2+\pi\mathbb Z+\pi\tau\mathbb Z$, while the denominator has simple zeros at $\pi\tau/2+\pi\mathbb Z+\pi\tau\mathbb Z$. The two sets are disjoint, so any period must lie in $\pi\mathbb Z+\pi\tau\mathbb Z$, by translating the <zero set>. The sign rule then gives the displayed exact lattice. Squaring the ratio removes that sign and has precisely the smaller lattice $\pi\mathbb Z+\pi\tau\mathbb Z$.
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