Unlocking the Riemann Hypothesis: Novel Pathways from Dynamical Systems

Exploring the Riemann Hypothesis through Dynamical Systems

This article proposes novel research pathways toward proving the Riemann Hypothesis (RH), inspired by a recent paper on dynamical systems and L-functions. The core idea is to leverage the mathematical structures presented in the paper to gain new insights into the behavior of the Riemann zeta function, ζ(s).

Framework 1: Exponential Moment Analysis

The paper introduces an exponential moment framework involving L-functions. We adapt this to the Riemann zeta function:

Formulation: gζ(s,z) = exp(iz/2 * ζ'(s)/ζ(s))

Potential Theorem 1: Bounds on the average value of |g_ζ(σ+it,z)| over vertical lines in the critical strip provide insights into the distribution of ζ(s)'s zeros.

Connection: The logarithmic derivative ζ'(s)/ζ(s) reveals the locations of zeros. Analyzing the moments of g_ζ(s,z) offers a new way to study these locations.

Framework 2: Average Value and Error Term Control

The paper analyzes average values and their associated error terms. We propose to extend this analysis to ζ(s):

Formulation: Let Δ(s,X) represent the error term in approximating averages of ζ(s) using a suitable method (e.g., based on the paper's techniques).

Potential Theorem 2: The behavior of Δ(s,X) as X → ∞ is directly related to the distribution of zeros of ζ(s) in the critical strip.

Connection: Precise control over error terms in approximations of ζ(s) can yield information about its zero distribution.

Framework 3: Infinite Product Representations

The paper discusses infinite product representations of L-functions. We can explore analogous representations for ζ(s):

Formulation: Investigate the infinite product representation of ζ(s) and its relation to sums involving functions similar to those appearing in the paper (e.g., functions related to the distribution of prime numbers).

Potential Theorem 3: Properties of the infinite product representation of ζ(s) could be used to establish criteria for the location of its zeros.

Connection: The infinite product representation provides a different perspective on the multiplicative structure of ζ(s) and its zeros.

Novel Approach 1: Combining Exponential Moments and Average Values

Methodology: Combine the exponential moment framework (Framework 1) with the analysis of average values (Framework 2). Study the joint behavior of g_ζ(s,z) and the error term Δ(s,X) to gain a more comprehensive understanding of ζ(s).

Predictions: This approach might reveal new relationships between the density of zeros and the magnitude of the error terms in approximating ζ(s).

Limitations: The analysis requires sophisticated techniques for handling oscillatory behavior and ensuring uniform error control.

Research Agenda

Conjectures:

• Conjecture 1: Bounds on the average value of |g_ζ(σ+it,z)| are directly linked to the number of zeros of ζ(s) in a given region.
• Conjecture 2: The decay rate of Δ(s,X) as X → ∞ determines the distribution of zeros of ζ(s) in the critical strip.

Mathematical Tools: Complex analysis, analytic number theory, probability theory, and potentially techniques from dynamical systems.

Intermediate Results: Establishing explicit formulas for the moments of g_ζ(s,z) and obtaining sharp bounds on Δ(s,X) would be significant steps.

Sequence of Theorems: The research would proceed by first establishing results for simplified cases (e.g., restricted regions in the critical strip) before tackling the full problem.

Simplified Cases: Begin by analyzing the behavior of ζ(s) in regions close to the critical line, where the zeros are most densely packed.