M Theory Lesson 18

I am very grateful to kneemo for spotting this paper by Carlos Castro on the connection between the Riemann hypothesis and dual string scattering amplitudes. Castro works at a place in Atlanta with a great name: the Center for Theoretical Studies of Physical Systems.

The paper begins by looking at scattering amplitudes of the form

$\frac{\zeta (ik) \zeta (1 - ik)}{\zeta (1 + ik) \zeta (- ik)}$

which appear in studies of particles moving in the hyperbolic plane. Observe that the poles of this expression occur when $k_n = i (0.5 + i \rho_n)$, corresponding to the zeroes of the Riemann zeta function. Thus one considers complex values of $k_n$ and the question arises as to what physical interpretation the zeroes should have.

Castro starts by looking at the Veneziano four point dual string amplitude, expressed in terms of the beta function. The zeros of $\zeta$ on the critical line appear to correspond to the real poles of the tachyonic amplitude.

On another note, check out the coolest blog post ever, on Never Ending Books!