This article analyzes potential connections between the Riemann Hypothesis (RH) and mathematical frameworks presented in arXiv:hal-00978908. It formulates concrete research pathways leveraging finite fields, polynomials, and code theory to understand the distribution of prime numbers and the zeros of the Riemann zeta function.
The paper utilizes the representation of Boolean functions over F₂ using polynomials:
∀(x₁, x₂, ..., xₘ) ∈ F₂ᵐ, p(x₁, x₂, ..., xₘ) ≡ f(x₁β₁ + x₂β₂ + ... + xₘβₘ) (mod 2)
where p is a Boolean function, f is a polynomial over F₂, and (β₁, β₂, ..., βₘ) is a basis for a vector space L over F₂. A potential research direction involves establishing a theorem linking the algebraic properties of the polynomial f to the combinatorial properties of the Boolean function p.
Connection to Zeta Function: Encode information about prime numbers into a Boolean function p. Analyze the polynomial f representing p. Properties of the Riemann zeta function (e.g., the location of its zeros) might be reflected in the algebraic structure of f. Relate the coefficients of the polynomial f to the prime counting function π(x) or related functions.
The paper mentions a divisibility property of the number of solutions to polynomial equations over finite fields:
"For a polynomial f ∈ F₂[X₁, X₂, ..., Xₘ] of degree k, the number of solutions in F₂ᵐ is divisible by 2⌈m/k⌉-1, where ⌈r⌉ denotes the smallest integer greater than or equal to r."
Generalizing this divisibility result to other finite fields and exploring tighter bounds is a potential research direction. The interplay between the degree k and the number of variables m is a key area to investigate.
Connection to Zeta Function: Construct polynomials whose solutions encode information about the distribution of primes. The divisibility properties of the number of solutions might reveal constraints on the distribution of primes, potentially leading to insights about the location of zeta zeros.
The paper touches upon the weight distribution of codewords:
"codewords of weight w is equal to µ × n, w̄ denotes the congruence of w/8 modulo 4." and "the weights are divisible by 2θ-1"
Exploring the relationship between the algebraic structure of the code and the divisibility properties of the weights of its codewords is a promising avenue. Constructing families of codes with specific weight divisibility properties could be beneficial.
Connection to Zeta Function: There's a potential analogy between the zeros of the zeta function and the weights of codewords. The Riemann Hypothesis suggests a specific distribution of these "defects" (zeta zeros). By studying codes with highly structured weight distributions, we might gain insights into the structure that underlies the distribution of primes and zeta zeros. Relate the weight enumerator of a code to a function related to the Riemann zeta function.
Mathematical Foundation: Encode prime gaps into a Boolean function. Let gₙ = pₙ₊₁ - pₙ be the nth prime gap. Define a Boolean function p(x₁, x₂, ..., xₘ) = 1 if the number represented by (x₁, x₂, ..., xₘ) is equal to a prime gap gₙ for some n, and 0 otherwise. Represent p as a polynomial. Combine with the work of Goldston, Pintz, and Yildirim on small gaps between primes.
Methodology: Compute the polynomial f for a large set of prime gaps. Analyze the coefficients of f. Investigate the divisibility properties of the number of solutions to the polynomial equation f(x₁, x₂, ..., xₘ) = 0 over F₂ᵐ.
Predicted Properties: The coefficients of f might exhibit a certain "smoothness" if the Riemann Hypothesis is true.
Potential Limitations: The degree of the polynomial f could be very high. This can be addressed by considering approximations to p that are represented by lower-degree polynomials.
Mathematical Foundation: Construct a family of error-correcting codes with specific weight distributions. Define a function Z_C(s) (analogous to the Riemann zeta function) based on the weight enumerator of the code C. Link this with existing work on explicit formulas in prime number theory.
Methodology: Analyze the "zeros" of Z_C(s). Establish a "code-theoretic Riemann Hypothesis": Conjecture that the "zeros" of Z_C(s) all lie on a specific line or within a specific region in the complex plane. Prove this code-theoretic Riemann Hypothesis for specific families of codes.
Predicted Properties: If the code-theoretic Riemann Hypothesis holds, it would suggest that the structured weight distribution of the code imposes constraints on the "zeros" of Z_C(s).
Potential Limitations: The analogy between codes and zeta functions might be too weak. Overcome this by carefully choosing families of codes that have properties that are strongly related to the distribution of primes.
The Riemann Hypothesis is related to the efficient computation of the zeta function. The complexity of boolean circuits is related to the complexity of computing boolean functions. There might be a way to connect these two ideas.
Specific Conjecture: The Riemann Hypothesis is true if and only if there exists a polynomial-size Boolean circuit that can approximate the Riemann zeta function to a given degree of accuracy.
Computational Experiments: Design Boolean circuits to approximate the Riemann zeta function and test their efficiency. If the circuits are indeed polynomial-size, then this would provide computational evidence in favor of the Riemann Hypothesis.
Ergodic theory studies the statistical properties of dynamical systems. The distribution of prime numbers can be viewed as a dynamical system.
Specific Conjecture: The Riemann Hypothesis is true if and only if the distribution of prime numbers satisfies certain ergodic properties, such as mixing or unique ergodicity.
Computational Experiments: Study the ergodic properties of prime number distribution using computational methods. If the distribution exhibits the conjectured ergodic properties, then this would provide computational evidence in favor of the Riemann Hypothesis.
The frameworks and approaches outlined here, combining polynomial representations, finite field analysis, and code-theoretic analogies, offer novel research pathways towards understanding and potentially proving the Riemann Hypothesis. While challenges remain, these structured approaches provide concrete directions for future investigation.