Unlocking the Riemann Hypothesis: Novel Pathways from Dynamical Systems

Exploring the Riemann Hypothesis through Dynamical Systems

This research investigates potential connections between dynamical systems and the Riemann Hypothesis, drawing inspiration from recent advancements in the field. The focus will be on leveraging the unique properties of chaotic systems and stochastic processes to gain new insights into the distribution of the zeros of the Riemann zeta function.

Framework 1: Moment Bounds with Exponential Dampening

The paper introduces a moment bound incorporating exponential dampening:

H << X²∫_T-H^T+H|ζ(σ+iv)|^kdv(∫_T-H/2^T+H/2exp(-c₁e^|v-t|/3)dt) + X^-1H

Proposed Theorem 1.1: For σ = 1/2 + ε, where ε > 0 is arbitrarily small, there exists a constant C(k,ε) such that:

∫_T^T+H|ζ(1/2 + ε + it)|^kdt ≤ C(k,ε)H(log T)^k²/2exp(-c₁√(εlog T))

If this bound holds for σ = 1/2, it would strongly suggest the Riemann Hypothesis.

Framework 2: Dirichlet Series Convergence

The paper analyzes the convergence of a Dirichlet series:

<< Σ_m=1^∞Σ_{n=1, n<m}^∞d_k(m)d_k(n)(mn)^-σ(log(m/n))^-1

Proposed Theorem 1.2: For σ > 1/2, define:

ζ*(s) = Σ_n=1^∞μ(n)n^-s(log(n+1)/log n)^-1/2

ζ*(s) is zero-free for σ > 1/2 if and only if ζ(s) satisfies the Riemann Hypothesis.

Framework 3: Asymptotic Expansions

The paper provides asymptotic expansions for a function related to the Riemann zeta function:

f(T,n) = 2Tarsinh(√(πn/(2T))) + √(2πnT + π²n²) - π/4 = -π/4 + 2√(2πnT) + (1/6)√(2π³)n^3/2T^-1/2 + ...

Proposed Theorem 1.3: The error term in the prime number theorem, ψ(x) - x = Σ_ρ(x^ρ)/ρ + R(x), has an asymptotic expansion involving terms similar to f(T,n), and |R(x)| = O(x^1/2+ε) if and only if the Riemann Hypothesis holds.

Novel Approach 1: Exponentially Weighted Zero-Free Regions

This approach combines frameworks 1 and 3 to explore zero-free regions using exponentially weighted integrals. By defining an exponentially weighted zeta function and establishing moment bounds, we aim to prove that the existence of zeros off the critical line leads to a contradiction.

Novel Approach 2: Enhanced Explicit Formula via Moment Analysis

This approach uses moment integrals on the critical line to refine the explicit formula for ψ(x). The goal is to demonstrate that the moment bounds imply the non-existence of zeros off the critical line.

Tangential Connection 1: Probabilistic Model

The error terms in the paper can be interpreted probabilistically, leading to a conjecture relating the Riemann Hypothesis to the asymptotic behavior of these error terms.

Tangential Connection 2: Arithmetic Function Correlations

The paper's results on arithmetic function correlations can be extended to study the correlation between ζ(s) and Dirichlet L-functions, leading to a conjecture relating these correlations to the Riemann Hypothesis.

Research Agenda

This research agenda outlines a multi-phase approach, focusing on establishing refined moment bounds, analyzing zero-free regions, and ultimately proving a contradiction based on the assumption of zeros off the critical line. Computational experiments using simplified cases would validate the approach.