On the Riemann Hypothesis, complex scalings and logarithmic time reversal

Perelman, Carlos Castro

doi:10.1016/j.geomphys.2018.03.002

“…We notice that the poles and zeros coincide in the pole dynamics when the angular momentum of the particle j is real and the Hamiltonian is there Hermitian, confirming the results obtained with the first approach finding interesting correspondences to the requirements dictated by the Regge trajectories in a hyperbolic support, viz., that the angular momentum operator j for a Hermitian operator satisfying the requirements to satisfy the RH must be real-valued. Another interesting approach to the RH is revisited within the framework of the special properties of θ functions, and CT invariance [102], but for the moment it goes beyond the purpose of the present work.…”

Section: Discussionmentioning

confidence: 99%

Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis

Tamburini¹,

Licata²

2021

View full text Add to dashboard Cite

The Riemann Hypothesis states that the Riemann zeta function ζ(z) admits a set of “non-trivial” zeros that are complex numbers supposed to have real part 1/2. Their distribution on the complex plane is thought to be the key to determine the number of prime numbers before a given number. Hilbert and Pólya suggested that the Riemann Hypothesis could be solved through the mathematical tools of physics, finding a suitable Hermitian or unitary operator that describe classical or quantum systems, whose eigenvalues distribute like the zeros of ζ(z). A different approach is that of finding a correspondence between the distribution of the ζ(z) zeros and the poles of the scattering matrix S of a physical system. Our contribution is articulated in two parts: in the first we apply the infinite-components Majorana equation in a Rindler spacetime and compare the results with those obtained with a Dirac particle following the Hilbert-Pólya approach showing that the Majorana solution has a behavior similar to that of massless Dirac particles and finding a relationship between the zeros of zeta end the energy states. Then, we focus on the S-matrix approach describing the bosonic open string scattering for tachyonic states with the Majorana equation. Here we find that, thanks to the relationship between the angular momentum and energy/mass eigenvalues of the Majorana solution, one can explain the still unclear point for which the poles and zeros of the S-matrix of an ideal system that can satisfy the Riemann Hypothesis, exist always in pairs and are related via complex conjugation. As claimed in the literature, if this occurs and the claim is correct, then the Riemann Hypothesis could be in principle satisfied, tracing a route to a proof.

show abstract

“…[27] with PT-symmetric operators cast some doubts on the feasibility of the so-called Hilbert-Pólya approach, as discussed in the Appendix (III A). Another approach to the RH is revisited within the framework of the special properties of θ functions, and CT invariance [44].…”

Section: From Hermitian Hamiltonians To the S-matrix Methods For The Rhmentioning

confidence: 99%

Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis

Tamburini,

Licata

2021

Preprint

0

View full text Add to dashboard Cite

The Riemann Hypothesis states that the Riemann zeta function ζ(z) admits a set of "non-trivial" zeros that are complex numbers supposed to have real part 1/2. Their distribution on the complex plane is thought to be the key to determine the number of prime numbers before a given number. We analyze two approaches. In the first approach, suggested by Hilbert and Pólya, one has to find a suitable Hermitian or unitary operator whose eigenvalues distribute like the zeros of ζ(z).In the other approach one instead compares the distribution of the zeta zeros and the poles of the scattering matrix S of a system. We apply the infinite-components Majorana equation in a Rindler spacetime to both methods and then focus on the S-matrix approach describing the bosonic open string for tachyonic states. In this way we can explain the still unclear point for which the poles and zeros of the S-matrix overlaps the zeros of ζ(z) and exist always in pairs and related via complex conjugation. This occurs because of the relationship between the angular momentum and energy/mass eigenvalues of Majorana states and from the analysis of the dynamics of the poles of S. As shown in the literature, if this occurs, then the Riemann Hypothesis can in principle be satisfied.

show abstract

The Riemann hypothesis and tachyonic off-shell string scattering amplitudes

Perelman

¹

2022

View full text Add to dashboard Cite

The 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.

show abstract

On the Riemann Hypothesis, complex scalings and logarithmic time reversal

Cited by 3 publications

References 32 publications

Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis

Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis

Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis

The Riemann hypothesis and tachyonic off-shell string scattering amplitudes

Contact Info

Product

Resources

About