I found this formula attributed to Kronecker relating solutions of Pell equation to exponential sum:

$$ 4 + \sqrt{17} \approx \frac{2}{9} e^{(5/18) \pi \sqrt{17}} \text{ and } \frac{1}{\sqrt{5}}e^{(1/10) \pi \sqrt{85}}$$

I can run this through a calculator and find both estimates quite convincing:

$4 + \sqrt{17} = 8.12310\dots$

$ \frac{2}{9} e^{(5/18) \pi \sqrt{17}} = 8.117409\dots $

$ \frac{1}{\sqrt{5}}e^{(1/10) \pi \sqrt{85}} = 8.0985607\dots $

These are impressive but not out-of-this-world. They are related to pell equation $t^2 - 17u^2 = - 1$ supposedly. And we are supposed to estimate: $$ \log \Big(t + u \sqrt{17}\Big)$$

in terms of theta functions. Can any one supply the details here. I know similar problems involving $e^{\pi\sqrt{163}}$ that are related to the class number formula.

What is the name for the link to these approximations and Pell equation?

See also History of the theory of numbers, Vol 2 (Eugene Dickson) p 378

Grenzformel), and was regarded at the time as a crowning jewel of the theory of modular functions. You will enjoy reading about this in the last chapter of Andre Weil's book (Elliptic functions according to Eisenstein and Kronecker), or in Vladut's book on the Jugendtraum. The limit formula, giving close approximations of this type, was at the basis of the Gelfond-Linnik-Baker solution to Gauss's class number one problem. $\endgroup$ – Vesselin Dimitrov Dec 12 '16 at 16:20