Unraveling the Riemann Hypothesis: A Novel Approach from Prime Number Recurrences

Introduction

The Riemann Hypothesis (RH) remains one of the most significant unsolved problems in mathematics. This article explores potential connections between a recently discovered set of prime number recurrence relations (from an unnamed source) and the RH, proposing several research pathways that could lead to a proof. The core idea is that these recurrence relations, while initially numerical and combinatorial, might encode information about prime distribution that, when properly interpreted, relates to the analytic properties of the Riemann zeta function.

Mathematical Frameworks and Connections to the Riemann Hypothesis

The unnamed source offers several mathematical structures that could be applied to the Riemann Hypothesis:

Framework 1: Nested Recurrence Relations

The source defines sequences using nested recurrence relations involving the i-th prime number, p_i. For example, one such relation is:

x₁(4) = 4
x₁(i) = (p_i-2 - 4) × x₁(i-1) for i > 4
x₂(3) = 2
x₂(i) = (p_i-1 - 3) × x₂(i-1) + x₁(i) for i > 3
#SP(5,2) = 0
#SP(5,i) = (p_i - 2) × #SP(5,i-1) + x₂(i) for i > 2

These relations generate integer sequences whose growth and divisibility properties might reveal insights into prime distribution. A key research direction involves constructing generating functions or Dirichlet series from these sequences and analyzing their analytic properties (poles, zeros).

Framework 2: Tables of Divisibility

The source also provides tables showing the divisibility of specific integers by small primes. The divisibility patterns might reflect a form of "sieve residue system" whose structure could be linked to the Möbius function or the Liouville function, both deeply connected to the Riemann Hypothesis.

Framework 3: Modular Arithmetic with Prime Offsets

The source explores recurrence relations with varying prime indices and subtraction constants. The sensitivity of sequence properties to these variations could be explored to find a "resonant" parameter set that might amplify or isolate aspects of prime distribution relevant to the RH.

Novel Approaches

Approach 1: Spectral Analysis of Recurrence-Generated Sequences

This approach proposes transforming the integer sequences generated by the recurrence relations into matrices or operators. The eigenvalues of these operators could potentially be linked to the zeros of the Riemann zeta function, aligning with the Hilbert-Pólya conjecture.

Approach 2: Analyzing Divisibility Patterns as a Sieve

This approach focuses on the divisibility tables. The patterns of divisibility by small primes could be analyzed as a type of sieve. The behavior of the sieve residues could be rigorously compared to known properties of functions related to the Riemann zeta function, such as the Möbius function. This comparison could provide new constraints on the location of the zeros.

Tangential Connections

While the source material is primarily numerical, tangential connections to established areas of research could provide valuable insights:

Connection 1: Additive Combinatorics

The patterns in the source's tables might reveal links to additive combinatorics. The additive energy of sets formed from the generated sequences could be analyzed to find potential correlations with the Riemann Hypothesis.

Connection 2: Prime Number Theorem Error Term

The recurrence relations could be interpreted as discrete analogues of the prime counting function. A detailed analysis could reveal how the error term in the prime number theorem is encoded in these relations, providing a path towards proving RH.

Research Agenda

The research agenda would involve several phases:

Phase 1: Foundation Building: Establish the asymptotic behavior and divisibility properties of the recurrence-generated sequences.
Phase 2: Connection Establishment: Formally connect the properties of the sequences to the distribution of primes and the error term in the prime number theorem.
Phase 3: Novel Proof Strategy: Develop a rigorous mathematical framework linking the properties of the sequences (or their transforms) to the Riemann Hypothesis.

Computational experiments would be crucial throughout the research, focusing on simplified cases and gradually increasing the complexity to validate the theoretical findings.