Mathematical Frameworks for the Riemann Hypothesis
The research presented in hal-00084606 introduces Scheduling by Edge Reversal (SMER) as a dynamical system for prime sieving. This framework provides several pathways for analyzing the distribution of primes and the behavior of the zeta function by examining how local node interactions generate global arithmetic coherence.
Framework 1: Coprimality Detection as a Dynamical Observable
A key finding in the SMER dynamics is the static arc phenomenon. When the number of arcs between nodes i and j is set to ri + rj - 1, a static arc appears if and only if the nodes share a common factor. This creates a distributed indicator for the greatest common divisor (gcd).
- Formulation: For random-like sampling of integer pairs, the average presence of static arcs is related to the value 1 - 1/zeta(2).
- Significance: This establishes a direct link between graph-based state evolution and the Euler product of the Riemann zeta function.
Framework 2: Periodicity and Spectral Gaps
The SMER orbit is characterized by a period P(i,j) determined by the least common multiple of node rates. As the system scales, the aggregate behavior of these periods across a primality-weighted graph mirrors the fluctuations in the prime counting function.
- Proposed Theorem: The Riemann Hypothesis is equivalent to the condition that the spectral gap of the SMER transition operator ensures square-root cancellation in the error terms of the prime counting function.
Novel Research Approaches
Approach 1: SMER-Wheel Discrepancy and Moebius Cancellation
This approach combines the SMER dynamics with wheel-sieve survivors. By analyzing the error term in the survivor count W(x;y), researchers can investigate the cancellation properties of the Moebius function. The goal is to prove that the SMER-induced permutation of residues satisfies a low-discrepancy condition across the network nodes.
Approach 2: Thermodynamic Formalism of Distributed Sieves
Treating the SMER system as a statistical mechanical model allows the definition of a partition function Z(s). If the potential function phi(n) = n + sqrt(n) - T(n) exhibits certain average-decrease properties as suggested in the source paper, it could demonstrate the absence of phase transitions for the real part of s greater than 1/2.
Tangential Connections and Agenda
The oscillatory peaks of the potential function phi(xi) at specific values suggest a resonance with the zeros of the zeta function. A detailed research agenda based on hal-00084606 involves:
- Conjecture: The spacing of SMER periods in a fully connected sieve graph follows Gaussian Unitary Ensemble statistics, mirroring the Montgomery-Odlyzko Law of zeta zero spacing.
- Experimental Validation: Simulate the SMER update rule on large-scale multigraphs to measure the error magnitude in prime counts relative to the Prime Number Theorem.
- Theoretical Goal: Establish a sequence of theorems linking the synchronization of edge reversals to the zero-free region of the zeta function on the critical line.