The Riemann Hypothesis and Tachyonic Off-Shell String Scattering Amplitudes

The study of the ${\bf 4}$-tachyon off-shell string scattering amplitude $ A_4 (s, t, u) $, based on Witten's open string field theory, reveals the existence of a continuum of poles in the $s$-channel and corresponding to a continuum of complex spins $ J $. The latter spins $ J$ belong to the Regge trajectories in the $ t, u$ channels which are defined by $ - J (t) = - 1 - { 1\over 2 } t = \beta (t)= { 1\over 2 } + i \lambda $; $ - J (u) = - 1 - { 1\over 2 } u = \gamma (u) = { 1\over 2 } - i \lambda $, with $ \lambda = real$. These values of $ \beta ( t ), \gamma (u) $ given by ${ 1\over 2 } \pm i \lambda $, respectively, coincide precisely with the location of the critical line of nontrivial Riemann zeta zeros $ \zeta (z_n = { 1\over 2 } \pm i \lambda_n) = 0$. We proceed to prove that if there were nontrivial zeta zeros (violating the Riemann Hypothesis) outside the critical line $ Real~ z = 1/2 $ (but inside the critical strip) these putative zeros $ don't$ correspond to any $poles$ of the ${\bf 4}$-tachyon off-shell string scattering amplitude $ A_4 ( s, t , u ) $. One of the most salient features of these results is the $collinearity$ of the ${\bf 4}$ off-shell tachyons. We may speculate that this spatial $collinearity$ is actually reflected in the $collinearity$ of the poles of the string amplitude, lying in the critical line : $ \beta = \gamma^* = { 1\over 2 } + i \lambda$, where the nontrivial zeta zeros are located. We finalize with some concluding remarks on continuous spins, non-commutative geometry and other relevant topics.