Exploring Prime Distribution Patterns for Clues to the Riemann Hypothesis

Recent mathematical explorations into prime number distributions offer intriguing connections that may bear relevance to the long-standing Riemann Hypothesis. A paper, arXiv:hal-02549967, introduces structured methods for analyzing integers based on their prime factors, suggesting new angles from which to view the fundamental properties of numbers.

Sieve Methods and Number Distribution

The paper describes sieve processes, akin to the Sieve of Eratosthenes but with specific stages (like 'Stage 2 - MeS' and 'Stage 2 - SuS'), for filtering natural numbers. These methods define sequences by removing numbers based on their divisibility by primes. For example, one step involves removing even numbers, leaving approximately 1/2 of the original set, followed by removing numbers defined by a 'thread' related to the prime 3, leaving a fraction of the remaining numbers.

These sieves provide a structured way to study the density of number sets that remain after removing multiples of initial primes. One approximation presented relates to the distribution of twin primes or similar pairs, highlighting how sieve outputs can approximate known constants in prime number theory.

The connection to the Riemann Hypothesis lies in its deep relationship with the distribution of prime numbers. The behavior of the Riemann zeta function, particularly the location of its non-trivial zeros, is dictated by how primes are spaced along the number line. If these sieve methods reveal fundamental, undiscovered regularities or patterns in prime distribution, they could offer new insights into the zeta function's properties.

Prime Power Representations and Threads

The paper also explores representing numbers using their prime power components, visualized as 'threads'. For instance, the thread for 2 is the sequence of powers [2⁰, 2¹, 2², 2³, ...], for 3 it is [3⁰, 3¹, 3², 3³, ...], and so on for every prime.

This representation is intrinsically linked to the Euler product formula for the Riemann zeta function, which expresses the function as an infinite product over prime numbers. Each term in this product corresponds to a geometric series involving powers of a single prime:

• The term for prime p is 1 / (1 - p^-s).
• Expanding this gives 1 + p^-s + p^-2s + p^-3s + ...
• This series involves the powers of p, directly mirroring the structure suggested by the prime 'threads'.

Analyzing the zeta function through the lens of these individual prime power 'threads' could potentially decompose its complex behavior into simpler, more manageable components. Understanding how the convergence and properties of each thread's series contribute to the overall function in the critical strip might reveal why the non-trivial zeros are conjectured to lie on the critical line.

Novel Research Pathways

Combining these frameworks suggests several research directions:

Sieve-Modified Zeta Functions

Define variants of the zeta function based on the sets of numbers remaining after specific sieve operations. Instead of summing over all integers n, sum only over the integers in the sieve-reduced set S:

ζ_S(s) = Σ_{x ∈ S} x^-s

By studying the analytic properties and zero distribution of ζ_S(s) for different sieve processes S, researchers could investigate how the inclusion or exclusion of certain number patterns affects the location of zeros. A key prediction is that if the sieve accurately models prime distribution, the zeros of the modified function might still lie on the critical line, or their deviation could provide clues about the necessary conditions for the Riemann Hypothesis.

Thread Operator Analysis

Formalize the concept of prime threads into mathematical operators. The logarithm of the zeta function can be expressed as a sum over contributions from each prime thread. Define operators that act on these individual thread contributions or analyze the properties of the threads themselves in relation to the complex variable s.

Investigating the spectral properties (eigenvalues and eigenfunctions) of such operators could potentially provide a new characterization of the zeta function's behavior. If the non-trivial zeros of zeta correspond to the eigenvalues of a specific operator derived from these prime threads, proving self-adjointness or other properties of this operator could lead directly to the Riemann Hypothesis.

Tangential Connections

The structured nature of sieves and prime threads hints at connections to other mathematical fields:

Dynamical Systems

The iterative application of sieve rules can be viewed as a discrete dynamical system. Each step transforms a set of numbers into a new set. The set of primes could potentially be seen as a fixed point or an invariant set of this system. Conjectures could be formulated linking the chaotic or ergodic properties of this dynamical system to the statistical distribution of primes and, by extension, to the Riemann Hypothesis.

Quantum Chaos

The distribution of energy levels in certain quantum chaotic systems is statistically similar to the distribution of the non-trivial zeros of the Riemann zeta function. The sieve processes or thread structures might provide a blueprint for constructing a mathematical operator (perhaps a Hamiltonian) whose spectrum corresponds to these zeros. Computational experiments could involve building and analyzing the spectral statistics of operators derived from these number-theoretic structures.

Detailed Research Agenda

A research program building on these ideas could proceed as follows:

Phase 1: Characterize Sieve Properties

• Conjecture: The asymptotic density of numbers remaining after applying a specific sequence of prime-based sieve operations is precisely predictable and relates to the prime number theorem error term.
• Tools: Combinatorics, number theory, asymptotic analysis, computational simulations.
• Intermediate Result: Formulas for the number of elements remaining in sieve-reduced sets for large N; quantitative comparison with known prime distribution functions.

Phase 2: Analyze Thread Contributions to Zeta

• Conjecture: The convergence properties of the sum of prime thread contributions to log(zeta(s)) in the critical strip are necessary and sufficient for the zeros to lie on the critical line.
• Tools: Complex analysis, functional analysis, operator theory.
• Intermediate Result: Detailed understanding of the analytic behavior of individual thread series; identification of conditions for convergence of their sum.

Phase 3: Develop Sieve-Modified Zeta Theory

• Conjecture: The non-trivial zeros of sieve-modified zeta functions ζ_S(s) lie on the critical line Re(s) = 1/2 if and only if the sieve S generates primes according to the distribution predicted by the Riemann Hypothesis.
• Tools: Analytic number theory, complex analysis, numerical computation of zeta zeros.
• Intermediate Result: Computation of zeros for ζ_S(s) for various sieves and parameters; analysis of their proximity to the critical line.
• Logical Sequence: Prove analytic continuation of ζ_S(s); establish a relationship between the properties of S and the zero-free regions of ζ_S(s).

Phase 4: Explore Operator & Dynamical Connections

• Conjecture: An operator derived from the prime thread representation exists whose eigenvalues correspond exactly to the non-trivial zeros of the Riemann zeta function.
• Tools: Operator theory, spectral geometry, quantum mechanics formalism.
• Intermediate Result: Construction of candidate operators; computation and analysis of their spectra.
• Logical Sequence: Define the operator formally; prove its relevant properties (e.g., self-adjointness); demonstrate the correspondence between its spectrum and zeta zeros.

This agenda outlines a path using the structures introduced in arXiv:hal-02549967 to build new theoretical and computational tools for probing the mysteries of prime numbers and the Riemann Hypothesis.