A well-known formula for the logarithm is given by


$$\log x = -\frac{\pi}{\operatorname{AGM}(a^2,b^2)}, \qquad x < 1$$


where AGM is the arithmetic-geometric mean, and $a$ and $b$ are given by


$$a = \sum\_{k\in\mathbb{Z}}x^{k^2}, \qquad
b = \sum\_{k\in\mathbb{Z}}x^{(k+1/2)^2};$$


Or, equivalently,


$$a = 1 + 2x + 2x^4 + 2x^9 + 2x^{16} + 2x^{25} + \dots$$
and
$$b = 2x^{1/4} + 2x^{9/4} + 2x^{25/4} + 2x^{49/4} + \dots$$


Why does it happen that $a = b$ occurs when $x = 0.8$?



Numerical coincidence? Why is $\sum(x^{k^2}) = \sum(x^{(k+1/2)^2})$ for $x = 0.8$?
The two values $a(0.8)$ and $b(0.8)$ appear "deceivingly" equal, but they actually are not!


Other "near-miss" values include
$$0<\theta\_3(0,0.9)-\theta\_2(0,0.9)<0.5\times 10^{-39}.$$


Let $f(x)=\theta\_3(0,x)-\theta\_2(0,x)$. The graph of $f(x)$, for values $0<x<1$ shows a global minimum at $x\_\*$ near $x=0.9$ (of course $f(x\_\*)>0$, still) and also a local maximum for some $0.9<x^\*<1$. It would be really interesting to figure out these numbers, especially $x\_\*$. In any case, there are two values of $x$ in the range $0.8<x<1$ for which
$$\theta\_3'(0,x)=\theta\_2'(0,x).$$


By the way, can someone post the graph for $y=f(x)$? It would be a nice documentation for the discussion and analysis here.


