Exploring Novel Mathematical Structures for the Riemann Hypothesis

Mathematical Frameworks and the Riemann Hypothesis

Based on the analysis of a recent paper (hal:03049500), several mathematical frameworks are identified as potentially relevant to the Riemann Hypothesis (RH). These include structured matrices involving factorial-like terms, bounds on algebraic number coefficients, polynomial division in finite fields, and algorithmic complexity analysis.

Polynomial Structures and Root Finding

The paper presents a matrix structure related to polynomial interpolation and root finding, particularly involving derivatives evaluated at multiple points. This suggests exploring connections between the roots of certain polynomials and the non-trivial zeros of the Riemann zeta function. If a polynomial could be constructed whose roots approximate or are directly related to these zeros, analyzing its properties (coefficient bounds, root distribution) could offer insights.

Bounds on Algebraic Numbers

Explicit bounds on the maximum absolute values of coefficients in polynomials or related algebraic numbers are given. If the locations of the zeta function's zeros, or related quantities, can be expressed in terms of algebraic numbers, these bounds could constrain the possible locations of non-trivial zeros, potentially helping to prove they lie on the critical line.

Polynomial Operations in Finite Fields

The paper discusses polynomial division and evaluation in finite fields (GF(p)). Concepts from the Weil conjectures connect equations over finite fields to zeta functions. Studying properties of polynomial quotients in GF(p), where coefficients are related to number-theoretic functions like the prime counting function or the von Mangoldt function, might reveal structural properties linked to the distribution of prime numbers and, by extension, the zeros of the zeta function.

Algorithmic Complexity

The analysis of algorithmic complexity, particularly for polynomial operations, suggests exploring computational approaches to the RH. Developing algorithms to compute properties of the zeta function or its zeros with a complexity dependent on the truth of RH could provide a computational test or reveal underlying structures.

Novel Approaches and Tangential Connections

Polynomial Root Distributions and Zeta Zeros

A novel approach involves constructing sequences of polynomials whose roots approximate the non-trivial zeros of the zeta function. Analyzing the coefficients of these polynomials for recurrence relations and verifying if they satisfy bounds derived from the paper could provide a link. The distribution of these polynomial roots could then be compared to the expected distribution of zeta zeros on the critical line.

Polynomial Quotients and Prime Gaps

Another direction is to relate polynomial division in finite fields to the distribution of prime numbers. By constructing polynomials with coefficients tied to prime-counting functions, properties like the periodicity of their quotient in GF(p) might correlate with prime number gaps, offering an arithmetic perspective on the zero distribution.

Connections to Other Fields

• Error Correcting Codes: Polynomials over finite fields are key to error-correcting codes. Connecting the coefficients of zeta-related Dirichlet series to codewords in a Reed-Solomon code could link the code's minimum distance to properties of zeta zeros or prime gaps.
• Dynamical Systems: The iterative evaluation of polynomials suggests a link to dynamical systems. Defining a system where a polynomial update rule governs complex numbers might yield Julia sets whose fractal dimensions relate to the density of zeta zeros on the critical line.
• Random Matrix Theory: Previous research suggests a connection between the distribution of zeta zeros and eigenvalues of random matrices. While not explicit in the snippets, the matrix structures could potentially tie into this area.

Detailed Research Agenda

The overall goal is to forge concrete links between the identified polynomial and finite field structures and the Riemann Hypothesis.

Key Conjectures:

• Polynomial Root Conjecture: A sequence of polynomials exists whose roots converge to zeta zeros, satisfying specific coefficient recurrence relations and bounds, and exhibiting a specific root distribution along the critical line if RH is true.
• Polynomial Quotient Conjecture: Polynomials constructed from prime-counting functions over GF(p) have quotients whose properties (like periodicity) are directly related to prime number distribution and, consequently, zeta zero distribution.

Required Tools and Techniques:

• Complex analysis, algebraic number theory, finite field arithmetic.
• Symbolic computation and numerical analysis for polynomial and zeta function calculations.
• Techniques from dynamical systems, coding theory, or random matrix theory depending on the chosen connection.

Intermediate Results:

• Establishing recurrence relations for coefficients of zeta-related polynomials.
• Proving convergence of polynomial roots to zeta zeros.
• Demonstrating a correlation between polynomial quotient properties in GF(p) and prime distribution characteristics.

Logical Sequence of Theorems:

1. Prove the existence and recurrence for coefficients of polynomials approximating zeta zeros.
2. Prove the convergence of the roots of these polynomials to the actual zeta zeros.
3. Establish the link between properties of specific polynomial quotients over GF(p) and prime number distribution.
4. Prove that if RH is true, these polynomial quotient properties satisfy a specific condition.
5. Show that if RH is false, this condition on polynomial quotients is violated, or the root distribution of approximating polynomials deviates from the critical line.

Simplified Example:

For a simple case, construct low-degree polynomials (e.g., degree 2 or 3) whose roots approximate the first few zeta zeros. Analyze their coefficients and compare their root distribution patterns as degree increases to known properties near the critical line. Similarly, examine polynomial quotients in GF(2) or GF(3) using small sets of prime-related coefficients to look for simple periodic patterns.