Integrality Constraints and Prime Valuations
The research presented in arXiv:2008.11234 introduces a method for testing primality through the "Whole" operator, which identifies whether a given expression results in an integer. By examining the ratio of factorials to near-square numbers, such as (n^2)! / (n^2 + 1), we can establish a system of prime valuation inequalities. This framework suggests that if a near-square number is composite, its prime factors must satisfy specific constraints within the factorial expansion of its predecessor.
This approach connects to the Riemann Hypothesis by translating discrete integrality constraints into prime counting statements. Using Legendre's formula, the valuation of factorials is linked to sums over prime powers, which are governed by the zeros of the zeta function. Establishing a contradiction in these constraints for large values of n could reveal new zero-free regions in the critical strip.
Truncated Euler Products and the Basel Problem
A significant technique involves the insertion of infinite Euler products that collapse into the Basel identity, where the product over all primes of p^2 / (p^2 - 1) equals pi^2 / 6. This suggests a research pathway focused on truncated Euler products where the error terms are analyzed for their relationship to the nontrivial zeros of the zeta function.
- Framework Formulation: Define a generalized Basel function that extends the prime product to the critical strip.
- Analytical Connection: Relate the approximation error of these truncated products to the distribution of zeta zeros.
- Potential Theorem: Prove that if the truncation error remains below a specific threshold, no zeros can exist with a real part greater than 1/2.
The Gaussian Integer Connection
The focus on primes of the form n^2 + 1 in arXiv:2008.11234 and arXiv:0802.1453v1 leads to the study of Gaussian integers. Since n^2 + 1 is the norm of the Gaussian integer n + i, the distribution of these primes is tied to the Dedekind zeta function. This provides a formal bridge: any method that proves the infinitude of near-square primes must account for the splitting behavior of primes in the Gaussian field.
Wolfram Language Experiment: Table[IntegerQ[Factorial[n^2] / (n^2 + 1)], {n, 1, 50}]
Proposed Research Agenda
The following agenda utilizes integrality constraints to target the Riemann Hypothesis:
- Phase 1: Establish the convergence of factorial-weighted zeta products and verify their behavior near the critical line using numerical methods.
- Phase 2: Prove that hypothetical zeros off the critical line would force large oscillations in the Whole operator's output, creating a contradiction with transcendental constants like pi.
- Phase 3: Use Diophantine approximation to show that the density of zeros must be exactly 1/2 to maintain the integrality of associated arithmetic functions.