Unveiling the Riemann Hypothesis: A Probabilistic and Analytic Approach

Probabilistic Models and the Riemann Hypothesis

The paper establishes a connection between the Riemann Hypothesis and the probability that two independent random variables are coprime. This is expressed as:

P[gcd(X, Y) = 1] = 6/π² + 6β/π² + O(β^(3/2-ε))

This probabilistic framework suggests a novel approach to studying the Riemann Hypothesis. By analyzing coprimality probabilities under varying distributions and connecting them to the statistical properties of greatest common divisor (GCD) distributions, we can potentially gain insights into the distribution of prime numbers. Further, a deeper analysis of error terms in probabilistic number theory could yield new constraints on the zeta function.

Connecting Coprimality to Zeta Function Zeros

A key avenue of research involves exploring the relationship between the asymptotic behavior of the probability of coprimality and the location of zeros of the Riemann zeta function. The integral equation below, derived from the paper, is pivotal:

∫₀^∞ f(β)β^(s-1)dβ = (Γ(s)ζ(s-1) - ζ(s))/ζ(s)

This equation connects the probabilistic model (through f(β)) to the zeta function, suggesting that the asymptotic behavior of the probability of coprimality, as β approaches 0, could reveal new characteristics of the zeta function's zeros.

Complex Analysis and Gamma Function Bounds

The paper also presents a bound on the Gamma function:

sup_σ∈[1/2,3] |Γ(σ + it)| = O(|t|^(5/2)e^(-π|t|/2))

This bound can inform our understanding of the analytic continuation of the zeta function. The behavior of the Gamma function within the critical strip could provide constraints on the growth of the zeta function, potentially limiting the behavior of ζ(s) near critical zeros.

Functional Equations and Zeta Function Zeros

Further research should focus on leveraging the functional equation of the Riemann zeta function, which involves the Gamma function. By combining the bound on the Gamma function with the functional equation, we could derive new constraints on the growth and behavior of the zeta function, potentially leading to constraints on the location of its zeros.

Zeta Distribution and Prime Number Distribution

The paper introduces the Zeta distribution, providing a probabilistic model for prime number distribution. This framework allows for simulations and analysis of prime number fluctuations, offering a potential avenue to test hypotheses related to the non-trivial zeros of ζ(s).

Simulations and Hypothesis Testing

Computational experiments using the Zeta distribution can be conducted to simulate the distribution of primes and compare these simulations to the predictions of the Riemann Hypothesis. Discrepancies between simulation results and the hypothesis could highlight areas requiring further investigation.