The Riemann hypothesis and tachyonic off-shell string scattering amplitudes

Carlos Castro Perelman

doi:10.1140/epjc/s10052-022-10429-3

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2022) 82: 469
https://doi.org/10.1140/epjc/s10052-022-10429-3

Regular Article - Theoretical Physics

The Riemann hypothesis and tachyonic off-shell string scattering amplitudes

Carlos Castro Perelman¹^,2^a

¹ Ronin Institute, 127 Haddon Place, 07043, Montclair, NJ, USA
² Center for Theoretical Studies of Physical Systems, Clark Atlanta University, 30314, Atlanta, Georgia

^a perelmanc@hotmail.com

Received: 3 February 2022
Accepted: 15 May 2022
Published online: 22 May 2022

Abstract

The study of the $4$ $\mathbf{4}$ -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 $- J (t) = - 1 - \frac{1}{2} t = β (t) = \frac{1}{2} + i λ$ $- J (t) = - 1 - { 1\over 2 } t = \beta (t)= { 1\over 2 } + i \lambda$ ; $- J (u) = - 1 - \frac{1}{2} u = γ (u) = \frac{1}{2} - i λ$ $- J (u) = - 1 - { 1\over 2 } u = \gamma (u) = { 1\over 2 } - i \lambda$ , with $λ = r e a l$ $\lambda = real$ . These values of $β (t), γ (u)$ $\beta ( t ), \gamma (u)$ given by $\frac{1}{2} \pm i λ$ ${ 1\over 2 } \pm i \lambda$ , respectively, coincide precisely with the location of the critical line of nontrivial Riemann zeta zeros $ζ (z_{n} = \frac{1}{2} \pm i λ_{n}) = 0$ $\zeta (z_n = { 1\over 2 } \pm i \lambda _n) = 0$ . 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 $J (k_{1}) + J (k_{2}) = J (k_{3}) + J (k_{4}) = 0$ $$ J ( k_1 ) + J (k_2) = J ( k_3 ) + J ( k_4 ) = 0$$ 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 $R e a l z = 1 / 2$ $$ Real~ z = 1/2 $$ (but inside the critical strip) these putative zeros $d o n^{'} t$ $$ don't$$ correspond to any poles of the $4$ $\mathbf{4}$ -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 $z = 0 + i 2 π n$ $z = 0 + i 2 \pi n$ , continuous spins, non-commutative geometry and other relevant topics.

© The Author(s) 2022

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