Theory Update 115

As number 26 suggests, the Riemann zeta value $\zeta (-3)^{-1} = - 120$ is associated to the four dimensional quaternions, whereas $\zeta (-7)^{-1} = - 240$ is associated to the octonions. The three parallelizable spheres give the triplet $( \zeta(-1), \zeta(-3), \zeta(-7) )$. Under the functional switch $s \mapsto 1 - s$ of zeta arguments, we obtain the triplet $(2,4,8)$ of dimensions for the corresponding number fields. In categorical scattering theory, we often note that $\zeta (n)$ is associated to $n+3$ particles, giving the triplet $(5,7,11)$ of primes for the exceptional Galois groups.

In the moduli triplet of twistor dimension, the genus $2$ orbifold characteristic was $\zeta (-3)$. The number $120$ counts the vertices of the three dimensional permutoassociahedron, which is obtained by turning each vertex of the $24$ vertex permutohedron into a pentagon. Hence, $120 = 24 \times 5$. In the complex case we have the string regulator, $\zeta (-1)^{-1} = -12$, like the $12$ pentagons that tile the Riemann sphere for the complex moduli space $M_{0,4}$, or the $12$ sides of the planar permutoassociahedron.

And if my calculator hand is working, real arguments give nice real numbers like $\zeta (26) = 1.0000000149$.