Prime Geodesics in AdS₂ and the Logarithmic Length Spectrum

This work proposes a geometric framework connecting prime numbers, hyperbolic geometry, and dynamical systems through the structure of AdS₂ and the group PSL(2,R). The central idea is that prime number weights emerge naturally from geometric properties of orbit projections in the geodesic flow on AdS₂. Starting from the Cartan decomposition of PSL(2,R), the geodesic flow on AdS₂ can be understood in terms of expanding and contracting directions. When group orbits are projected onto the tangent space, this decomposition produces an intrinsic asymmetry between two directions: one corresponding to expansion and the other to contraction. Under the AdS₂/CFT₁ correspondence, these two directions can be interpreted physically as boundary time dilation and energy scaling. Because PSL(2,R) preserves a symplectic structure, the expansion and contraction factors are not independent. Instead, they obey a conservation relation in which the product of the time scaling and the energy scaling remains constant. This conservation law emerges as a direct geometric consequence of the group structure rather than as an external physical assumption. Within this framework, two constructions that were originally developed independently can be identified as different realizations of the same underlying Cartan element. The first construction is a dynamical model based on a delta-kick system. In this model, the Hessian of the action functional develops a stable eigenvalue structure organized by prime-indexed layers. The scaling behavior of these eigenvalues matches the contraction factor predicted by the Cartan decomposition. The second construction is a geometric model based on cone or torus monodromy. In this setting, the trace of the monodromy matrix associated with each prime layer matches precisely the trace of the same Cartan element that appears in the geodesic flow description. Although these two models originate from very different perspectives—one dynamical and one geometric—they converge to the same algebraic structure. This agreement provides a mutual justification for the Cartan projection framework. From this unified structure, the length of the irreducible closed geodesic associated with a prime number becomes logarithmic in the prime. As a consequence, the weight assigned to each prime orbit matches the factor that appears in analytic number theory. When this length spectrum is substituted into the Selberg zeta function associated with the hyperbolic geometry, the resulting product reproduces the structure of the Riemann zeta function. Rather than presenting a complete proof of the Riemann hypothesis, the purpose of this work is to identify a structural bridge linking three different perspectives: hyperbolic geometry of AdS₂, dynamical prime-layer systems, and analytic number theory through zeta functions. The paper clarifies which components of this structure are already established within the geometric framework and which elements remain open problems. In particular, the global realization of a surface whose prime geodesic spectrum exactly matches the logarithmic prime structure remains an open direction for future work.