The Riemann hypothesis and tachyonic off-shell string scattering amplitudes

The study of the 4-tachyon off-shell string scattering amplitude A(4)(s, t, u), based on Witten's open string field theory, reveals the existence of poles in the s-channel and associated to a continuum of complex "spins" J. The latter J belong to the Regge trajectories in the t, u channels which are defined by -J(t) = -1 - 1/2t = beta(t) = 1/2 + i lambda; -J(u) = -1 - 1/2u = gamma(u) = 1/2 - i lambda, with lambda = real. These values of beta(t), gamma(u) given by 1/2 +/- i lambda, respectively, coincide precisely with the location of the critical line of nontrivial Riemann zeta zeros zeta(z(n) = 1/2 +/- i lambda(n)) = 0. It is argued that despite assigning angular momentum (spin) values J to the off-shell mass values of the external off-shell tachyons along their Regge trajectories is not physically meaningful, their net zero-spin value J (k(1)) + J (k(2)) = J(k(3)) +J(k(4)) = 0 is physically meaningful because the on-shell tachyon exchanged in the s-channel has a physically well defined zero-spin. 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 4-tachyon off-shell string scattering amplitude A(4)(s, t, u). We finalize with some concluding remarks on the zeros of sinh(z) given by z = 0 + i pi n, continuous spins, non-commutative geometry and other relevant topics.