Unlocking the Riemann Hypothesis: Novel Pathways from Finite Field Extensions

Exploring Unconventional Paths to the Riemann Hypothesis

This article proposes several research pathways toward proving the Riemann Hypothesis (RH), inspired by a recent paper on finite field extensions and symmetric bilinear complexity (arXiv:1706.09139). The paper's focus on finite fields might seem unrelated to the RH, but we will show how its mathematical structures can be adapted and leveraged to study the zeta function.

Framework 1: Finite Field Extensions and Zeta Function Zeros

The paper explores extensions of finite fields λ_p² and λ_p. We propose adapting the inequality structures from the paper to analyze the distribution of zeros of the Riemann zeta function. The paper's bounds on symmetric bilinear complexity, particularly those involving ε_p(n), could provide insights into the distribution of zeta zeros.

Formulation: Define a modified zeta function, ζ_sym(s), incorporating the symmetric complexity function μ_p^sym(n) from the paper: σ_sym(s) = ∑ μ_p^sym(n)/n^s.
Potential Theorem: Prove that the distribution of zeros of σ_sym(s) is related to the distribution of zeros of the Riemann zeta function, σ(s).
Connection: Establish a formal link between the bounds on μ_p^sym(n) and the density of zeros of σ(s) on the critical line.

Framework 2: Epsilon Function and Prime Distribution

The paper's ε_p(n) function, defined as ε_p(n) = (2n/(p-3))^(α-1), exhibits structural similarities to error terms found in prime-counting functions. This connection could be exploited to refine existing error bounds related to the RH.

Formulation: Investigate the relationship between ε_p(n) and the error term in the prime number theorem, |π(x) - li(x)|.
Potential Theorem: Prove tighter bounds on |π(x) - li(x)| using properties of ε_p(log x).
Connection: Demonstrate how improved bounds on the prime distribution translate into refined estimates for the location of zeros of σ(s).

Framework 3: Functional Transformations and Gamma-Zeta Relations

The paper implicitly suggests a connection between finite fields and the distribution of primes. This can be explored by investigating functional equations involving gamma functions and the zeta function. Such transformations might reveal hidden symmetries of the zeta function, leading to new insights about its zeros.

Formulation: Explore functional equations of the form f(x) = g(u), where f(x) involves the zeta function and g(u) involves gamma functions and potentially aspects of finite field structures.
Potential Theorem: Show that the zeros of σ(s) are related to singularities of g(u) or symmetries in the transformation.
Connection: Establish a direct link between the zeros of the completed zeta function Ξ(z) and the zeros or singularities of the transformed function g(u).

Novel Approaches

Approach 1: Combining Symmetric Complexity and Explicit Formulas

We propose combining the symmetric complexity function from the paper with explicit formulas for σ(s) involving prime powers. This approach could reveal new relationships between the distribution of primes and the location of zeta zeros.

Approach 2: Field Extension Height Functions and the Critical Line

This approach focuses on establishing a connection between the field extension bounds from the paper and the height of points on the critical line L(1/2 + it). This might lead to a new understanding of the distribution of zeros on the critical line.

Tangential Connections

Further research could explore tangential connections between the paper's results and other areas of mathematics, such as random matrix theory or the theory of L-functions. The goal is to find formal mathematical bridges that connect seemingly unrelated concepts to the RH.

Research Agenda

A detailed research agenda would involve a phased approach, starting with proving explicit bounds for μ_p^sym(n), establishing the analytic properties of σ_sym(s), and then connecting these results to the distribution of zeros of σ(s). This would require a combination of analytic number theory techniques, finite field arithmetic, and potentially computational experiments to validate the proposed connections.