Unraveling the Riemann Hypothesis: Novel Insights from Prime Number Sieves

Sieve-Based Zero Prediction

The paper's advanced prime sieving methods offer a potential framework for analyzing the distribution of prime numbers, which is intrinsically linked to the non-trivial zeros of the Riemann zeta function, ζ(s). The sieve formulation, S_k+1 = W_k+1 \ {y × p_k+1 | y ∈ W_k+1}, provides a structured way to generate primes. We propose to investigate how the distribution of primes generated via this method correlates with the density and spacing of zeros along the critical line (Re(s) = 1/2).

Formulation: Develop a refined sieve algorithm that correlates prime gaps with intervals on the critical line.
Potential Theorem: Prove a statistical relationship between prime gaps generated by the sieve and the density of zeros in corresponding intervals on the critical line.
Connection: Establish criteria or bounds for zero locations based on the statistical properties of prime gaps.

Coprimality and Zero Spacing

The concept of coprimality, as discussed in the paper, could influence the regularity or irregularity in the spacing of zeros of ζ(s). This framework could reveal new patterns in zero distribution.

Formulation: Define a ‘coprimality density’ measure for intervals along the critical line, based on the coprimality relationships between numbers in the sieve.
Potential Theorem: Prove a correlation between this coprimality density and the spacing between consecutive zeros.
Connection: Establish a quantitative relationship between coprimality properties and the distribution of zeros.

Graph Theoretical Representations

Representing number sets using the multigraphs described in the paper may offer a novel way to visualize and analyze the distribution of zeta zeros. The connectivity and path properties within this graph could reveal inherent patterns.

Formulation: Construct a graph where vertices represent primes and edges represent coprimality or other relevant relationships between numbers generated by the sieve.
Potential Theorem: Prove that topological properties of this graph (e.g., clustering coefficient, path lengths) correlate with the distribution of zeta zeros.
Connection: Relate graph-theoretical measures to properties of the Riemann zeta function.

Research Agenda

This research requires a multi-pronged approach. Initially, we focus on establishing correlations between sieve outputs and zero distributions. Rigorous proofs of these correlations are essential. Then, we will develop and prove theorems connecting coprimality, sieve residues, and zero behavior. Advanced sieving techniques, analytic number theory tools, and computational simulations will be crucial for validating the proposed connections.