Theory Update 114

The other formula given by number 26 uses $\alpha$, the fine structure constant, starting from the observation that ${\alpha }^{-1}$ is close to $137$. Many have attempted derivations of the modern non integral value for $\alpha$ using the integer $137$. For instance, the Relativist James G. Gilson discusses the beautiful formula

$\alpha =\frac{\cos (\pi /137)}{137}$

noting that other authors have expressed $137$ as $2^7+2^3+2^0$, as used by number 26. The integer $163=137+26$ is a favourite Almost Integer known as Ramanujan's constant. It satisfies

$e^{\pi \sqrt{163}}\simeq 640320^3+744$

where the decomposition of the large integer is well known in moonshine circles. Alternatively, we can write

$e^{\pi \sqrt{163}}\simeq (C{e}^{\pi i/24}{)}^{24}-24$

for $C=\eta \left(z\right)/\eta \left(2z\right)$, where $z=(1+\sqrt{163}i)/2$ and $\eta$ is the Dedekind eta function. The $j$-invariant also appears in interesting studies of CFTs, with central charge $48$. Pundits call all this numerology, but mathematicians most certainly do not. Long live the bosonic string.