Unlocking Zeta Zeros: Prime Patterns and Recursive Structures

Exploring Novel Connections for the Riemann Hypothesis

New research in number theory, detailed in arXiv:1503.07688, presents mathematical structures involving alternating sums, divisibility constraints, and recursive sequences related to prime numbers. While not directly aimed at the Riemann Hypothesis (RH), these frameworks offer intriguing possibilities for developing novel approaches to this long-standing problem concerning the non-trivial zeros of the Riemann zeta function.

Key Mathematical Frameworks from the Paper

Alternating Sums and Divisibility Constraints

The paper utilizes formulas for integers R involving alternating signs and products of constants (often small primes) and variables subject to specific divisibility rules. A key structure is represented by sums like:

R = (-1)^b₁ * C₁ * k₁ + (-1)^b₂ * C₂ * k₂ + ...

where b_i determine the sign, C_i are constants, and k_i are integers such that k_i / C_i is not an integer. This framework suggests a structured way to generate or analyze numbers, potentially primes, based on modular and sign properties.

Modulo Arithmetic and Arithmetic Progressions

Constraints are imposed on variables using modulo arithmetic, excluding certain values based on arithmetic progressions. For example, variables K are restricted such that K is not equal to 7m + 3 for specific ranges of m. Such constraints are fundamental in number theory, particularly concerning the distribution of primes within arithmetic progressions, a topic closely related to Dirichlet L-functions.

Recursive Structures and Integer Sequences

The paper's use of tables and sequential definitions hints at underlying recursive relationships or the generation of specific integer sequences. Analyzing the properties of these sequences, such as their growth rates and distribution, could provide a new lens through which to view number theoretic functions.

Novel Approaches Combining Frameworks with RH Research

Constructing Zeta-Like Functions from Paper Structures

One approach is to define new functions analogous to the Riemann zeta function, but whose terms are derived from the structures in arXiv:1503.07688. For instance, one could define a series:

F(s) = ∑ a_n / n^s

where the coefficients a_n are constructed using the alternating sums and divisibility constraints, or are elements of the recursive sequences presented. The goal would be to analyze the analytic continuation of F(s) and the location of its zeros.

Proposed Methodology:

Define coefficients a_n based on specific interpretations of the paper's formulas, potentially linking them to prime number properties.
Study the convergence properties and attempt to find an analytic continuation for F(s).
Analyze the zero distribution of F(s).

Concrete Predictions: If the coefficients are chosen appropriately, the non-trivial zeros of F(s) might lie on the critical line, or their distribution might mirror that of the Riemann zeta function's zeros, providing insight into their structure.

Limitations: Establishing analytic continuation and proving the location of zeros for such a function F(s) is a significant challenge. The finite nature of some structures in the paper would need to be adapted to infinite series. Overcoming this requires careful construction of the coefficients and the application of advanced complex analysis techniques.

Linking Modulo Constraints to Spectral Properties

The modulo constraints and patterns in arithmetic progressions could potentially be linked to the spectral properties of certain operators or matrices, which are known to have connections to the distribution of zeta zeros through areas like quantum chaos and random matrix theory.

Proposed Methodology:

Construct a family of matrices or operators whose properties are encoded by the arithmetic progression constraints described in the paper.
Analyze the eigenvalue distribution of these mathematical objects.
Compare the statistical properties of these eigenvalues to the statistical properties of the non-trivial zeros of the zeta function.

Concrete Predictions: The statistical distribution of the eigenvalues might match the GUE (Gaussian Unitary Ensemble) distribution, which is conjectured to describe the spacing between consecutive non-trivial zeta zeros.

Limitations: Formally constructing such mathematical objects and proving the rigorous link between their spectra and the zeta function's zeros is highly complex. This requires deep knowledge of spectral theory, random matrix theory, and number theory. Initial steps could involve computational experiments to test correlations.

Tangential Connections

Ergodic Theory and Prime Patterns

The patterns and constraints on integers and primes presented in the paper could potentially be modeled using concepts from ergodic theory. Viewing the sequence of prime numbers or related integer sequences as a dynamical system allows for the study of their long-term average behavior and distribution properties.

Formal Bridge: The prime number theorem and related results describe the density and distribution of primes. The structures in arXiv:1503.07688 provide specific rules for constructing or selecting integers based on prime factors and modular conditions. An ergodic system could be designed where states represent properties derived from these rules, and the dynamics reflect transitions based on the sequence generation.

Conjecture: There exists a specific ergodic system whose statistical properties (e.g., mixing properties, spectral measure) are directly related to the distribution of prime numbers, and the Riemann Hypothesis is equivalent to a statement about the properties of this system's invariant measure or spectrum.

Computational Experiments: Simulate simplified versions of the proposed ergodic system based on the paper's rules and compare statistical outputs (like frequency distributions or correlations) with known data on prime numbers or zeta zero distributions.

Detailed Research Agenda

Overall Goal: Leverage the structural insights from arXiv:1503.07688 to construct a mathematical object (a function, an operator, a dynamical system) whose properties, particularly its zero/eigenvalue distribution, are equivalent to the Riemann Hypothesis.

Key Conjectures to be Proven:

Conjecture 1: A zeta-like function F(s), constructed with coefficients based on the alternating sum/divisibility constraints, has all its non-trivial zeros on the line Re(s) = 1/2.
Conjecture 2: The statistical distribution of eigenvalues of operators derived from the paper's modulo arithmetic constraints matches the GUE distribution.

Mathematical Tools and Techniques Required:

Complex Analysis (analytic continuation, functional equations, contour integration)
Analytic Number Theory (prime number theorem, sieve methods, Dirichlet series)
Spectral Theory (properties of operators and matrices)
Probability and Statistics (random matrix theory, statistical analysis of distributions)
Ergodic Theory (dynamical systems, invariant measures)

Potential Intermediate Results:

Proof of analytic continuation for the proposed function F(s).
Derivation of a functional equation for F(s).
Establishment of bounds on the zero-free regions for F(s) or the eigenvalue distribution of the derived operators.
Demonstration that the statistical properties of derived sequences or distributions align with predictions from number theory or random matrix theory for small cases.

Logical Sequence of Theorems:

Theorem A: Define the function F(s) and prove its region of convergence.
Theorem B: Prove the analytic continuation of F(s) to the complex plane.
Theorem C: Establish a functional equation for F(s).
Theorem D: Prove that all non-trivial zeros of F(s) lie on the critical line Re(s) = 1/2 (This would imply RH if F(s) is shown to be equivalent to ζ(s)).

Alternatively, for the spectral approach:

Theorem A': Define the family of operators/matrices based on the paper's constraints.
Theorem B': Prove properties of the spectra of these objects.
Theorem C': Rigorously demonstrate that the statistical distribution of these spectra is identical to that of the non-trivial zeros of ζ(s).

Explicit Examples:

Simplified F(s): Construct a series using a simple instance of the alternating sum formula, e.g., a_n = (-1)^b(n) * C(n) * k(n) where b(n), C(n), k(n) are simple functions of n derived from the paper's patterns. Analyze the analytic properties of ∑ a_n / n^s.
Modulo Constraint Matrix: Create a small matrix (e.g., 5x5) where entries depend on satisfying or violating a modulo constraint from the paper. Compute its eigenvalues and analyze their distribution.