The Riemann hypothesis and tachyonic off-shell string scattering amplitudes

Abstract

AbstractThe study of the $$\mathbf{4}$$ 4 -tachyon off-shell string scattering amplitude $$ A_4 (s, t, u) $$ 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 $$ - J (t) = - 1 - { 1\over 2 } t = \beta (t)= { 1\over 2 } + i \lambda $$ - J ( t ) = - 1 - 1 2 t = β ( t ) = 1 2 + i λ ; $$ - J (u) = - 1 - { 1\over 2 } u = \gamma (u) = { 1\over 2 } - i \lambda $$ - J ( u ) = - 1 - 1 2 u = γ ( u ) = 1 2 - i λ , with $$ \lambda = real$$ λ = r e a l . These values of $$ \beta ( t ), \gamma (u) $$ β ( t ) , γ ( u ) given by $${ 1\over 2 } \pm i \lambda $$ 1 2 ± i λ , respectively, coincide precisely with the location of the critical line of nontrivial Riemann zeta zeros $$ \zeta (z_n = { 1\over 2 } \pm i \lambda _n) = 0$$ ζ ( z n = 1 2 ± i λ 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 $$ J ( k_1 ) + J (k_2) = J ( k_3 ) + J ( k_4 ) = 0$$ 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 $$ Real~ z = 1/2 $$ R e a l z = 1 / 2 (but inside the critical strip) these putative zeros $$ don't$$ d o n ′ t correspond to any poles of the $$\mathbf{4}$$ 4 -tachyon off-shell string scattering amplitude $$ A_4 (s, t, u) $$ A 4 ( s , t , u ) . We finalize with some concluding remarks on the zeros of sinh(z) given by $$ z = 0 + i 2 \pi n$$ z = 0 + i 2 π n , continuous spins, non-commutative geometry and other relevant topics.