Unveiling the Riemann Hypothesis: A Novel Approach Through Dynamical Systems

Exploring Prime Number Dynamics and the Riemann Hypothesis

This research investigates a novel pathway towards proving the Riemann Hypothesis by analyzing the dynamics of prime number distribution and their connection to the Riemann zeta function. The approach draws inspiration from recent work on prime number distribution and applies these insights to the properties of the zeta function.

Framework 1: Logarithmic Inequalities

The core of this approach lies in the analysis of logarithmic inequalities related to the distribution of prime numbers. A key inequality from the source material (arXiv:XXXX.XXXXX) is:

log log θ(qn′) - log log θ(qn) ≥ Σ(qn

where θ(x) is the Chebyshev function and qn represents the nth prime number. This inequality provides bounds on the gaps between consecutive primes. The central conjecture is that these bounds, when extended and refined, can be linked to the distribution of zeros of the Riemann zeta function.

Framework 2: Chebyshev's Function and Zeta Function Zeros

Chebyshev's function, θ(x), plays a crucial role in this approach. We aim to establish a strong connection between the asymptotic behavior of θ(x) and the location of zeta function zeros. A key lemma to be developed is:

Lemma 2.1: If for all x ≥ 2, θ(x) = x + O(√x log²x), then the Riemann Hypothesis holds.

This lemma bridges the asymptotic behavior of prime distribution with the properties of the zeta function.

Framework 3: Prime Product Analysis

The infinite product representation of the Riemann zeta function offers another avenue for investigation. We explore the implications of inequalities such as:

θ(qn)^(1/qn) ≥ 1 + log(qn+1)/θ(qn)

This inequality, combined with the properties of infinite products, could provide constraints on the possible locations of zeta zeros. We will investigate the convergence properties of these prime products and their relationship to the Euler product representation of the zeta function.

Novel Approach: Combining Frameworks

A novel approach combines the logarithmic inequalities and Chebyshev's function analysis. The central theorem is:

Theorem 3.1: If for all n ≥ 1, the logarithmic inequality holds and θ(x) = x + O(√x log²x) for all x ≥ 2, then the Riemann Hypothesis holds.

This theorem provides a direct path from the dynamics of prime number distribution to the Riemann Hypothesis. The proof will involve establishing a precise relationship between the bounds on prime gaps and the location of zeta zeros.

Research Agenda

• Conjecture 1: The logarithmic inequality implies a specific growth rate for θ(x) that is sufficient to prove the Riemann Hypothesis.
• Conjecture 2: The prime gaps satisfy stronger bounds than currently known, directly impacting the distribution of zeta zeros.

The research will involve advanced techniques from prime number theory and complex analysis, focusing on establishing the connections between the logarithmic inequalities, Chebyshev's function, and the Riemann zeta function. Computational experiments will be conducted to validate the conjectures and provide numerical evidence supporting the Riemann Hypothesis.