Let $q\in (0,1)$. Is there a way of computing the series $ \sum_{n=0}^\infty q^{n^2} $ explicitly? Is there at least a nice accurate estimate?
All I could get is the estimate $\sqrt{\frac{\pi}{4\cdot\mathrm{ln}\frac{1}{q}}}\leq\sum_{n=0}^\infty q^{n^2}\leq 1+\sqrt{\frac{\pi}{4\cdot\mathrm{ln}\frac{1}{q}}}$ via integration (quite possibly flawed).
For $q=\frac{1}{2}$, Maple gives the values $ 1.064467020\leq 1.564468414\leq 2.064467020, $ showing that my estimate is not very precise. (Of course, the sum of the two errors will always be $1$. Here both errors are coincidentally almost exactly $\frac{1}{2}$.)