The Riemann hypothesis and tachyonic off-shell string scattering amplitudes

Abstract

The study of the mathbf{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 - { 1over 2 } t = beta (t)= { 1over 2 } + i lambda ; - J (u) = - 1 - { 1over 2 } u = gamma (u) = { 1over 2 } - i lambda , with lambda = real. These values of beta ( t ), gamma (u) given by { 1over 2 } pm i lambda , respectively, coincide precisely with the location of the critical line of nontrivial Riemann zeta zeros zeta (z_n = { 1over 2 } pm 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 mathbf{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 2 pi n, continuous spins, non-commutative geometry and other relevant topics.