We are familiar with the expansion of the j-function,


$$j(\tau) = \tfrac{1}{q}+744+ 196884{q} + 21493760{q}^2 + \dots\tag1$$


and maybe with the approximation,


$$e^{\pi\sqrt{652}} = (640320^3+744)^2-2\cdot196883.999999999918\dots$$


but can somebody give a *short*, non-specialist explanation (if it is even possible), why the relationship $a\approx b$ between,


$$\begin{aligned}
\log(196883)\; &\approx 12.19 = a\\
4\pi\; &\approx 12.56 = b
\end{aligned}$$


is suddenly mentioned, of all places, in [quantum gravity](https://en.wikipedia.org/wiki/Monstrous_moonshine#Conjectured_relationship_with_quantum_gravity)? (Witten's paper is [here](http://arxiv.org/pdf/0706.3359.pdf).)




---


$\color{brown}{Edit:}$ (To address possible comments)


Witten defines a certain function $Z\_k(q)$ in page 30, and for the first few $k$,


$$\begin{aligned}
J(q) = Z\_1(q) &= q^{-1}+196884q+\dots\\
Z\_2(q) &= q^{-2}+1+42987520q+\dots\\
Z\_3(q) &= q^{-3}+q^{-1}+1+2593096794q+\dots\\
Z\_4(q) &= q^{-4}+q^{-2}+q^{-1}+2+81026609428q+\dots
\end{aligned}$$


On a hunch, I used *Mathematica's* Integer Relations and checked these coefficients with the coefficients $c\_n>1$ of $J(q)$,


$$c\_n =196884, 21493760, 864299970, 20245856256,\dots$$


([OEIS A014708](https://oeis.org/A014708)) and, sure enough, they were just simple linear combinations,


$$\begin{aligned}
196884\; &=c\_1\\
42987520\; &= 2c\_2\\
2593096794\; &= c\_1+3c\_3\\
81026609428\; &= c\_1+2c\_2+4c\_4
\end{aligned}$$


Using the general formula at the bottom of p.34,


$$\begin{aligned}
\log(c\_1)\;&\approx 12.19\\
4\pi\sqrt{1}\; &\approx 12.56\\[2.5mm]
\log(2c\_2)\;&\approx 17.57\\
4\pi\sqrt{2}\; &\approx 17.77\\[2.5mm]
\log(c\_1+3c\_3)\;&\approx 21.67\\
4\pi\sqrt{3}\; &\approx 21.76\\[2.5mm]
\log(c\_1+2c\_2+4c\_4)\;&\approx 25.12\quad\quad\quad\quad\\
4\pi\sqrt{4}\; &\approx 25.13\\
\end{aligned}$$


and the paper states that ***"...agreement improves rapidly if one increases k..."*** for the Bekenstein-Hawking entropy. (*Whatever that is.)*



On $e^{\pi\sqrt{4\cdot163}}$ and unusual connections
I'm not an expert on black holes, but I can give you a couple pointers. From work of Bekenstein and Hawking in the 1970s, we are pretty sure that macroscopic black holes in our 3+1 dimensional universe behave like thermodynamic objects. They have temperature, and entropy (reflecting some hidden microstates), and the entropy is proportional to the surface area of the event horizon. Since the derivation of their formula did not use quantum gravity, one expects that quantum corrections become relevant when one considers very small black holes.


The black holes that appear in this question live in $AdS\_3$ space, which is a 2+1 dimensional spacetime that has $SL\_2(\mathbb{R})$-geometry (which is kind of negatively curved). While this universe is quite different from our own, one obtains an entropy versus surface area relationship for black holes by similar reasoning, and again one expects some quantum corrections to show up in the small entropy regime. Explicit black hole solutions to Einstein's equations were found by Bañados, Teitelboim, and Zanelli, and they were found to have event horizons with positive surface area (which is really circumference when we consider 2+1 dimensions). 


When quantum gravity is brought into the picture, the sizes of possible black holes become quantized. Following AdS/CFT, Witten conjectures that size corresponds to conformal weight of a primary field, and this is why you see the formula $4\pi\sqrt{k}$.


The near-integer behavior of $e^{2\pi \sqrt{163}}$ does not seem particularly connected to any of this. It is basic class field theory, with some Hecke operators. See Chapter 3 (I think) of Silverman's *Advanced Topics* and [this MathOverflow question](https://mathoverflow.net/questions/4775/why-are-powers-of-exp-pi-sqrt163-almost-integers).


