Prime Number Distribution and the Riemann Hypothesis

Exploring the Riemann Hypothesis through Prime Number Distribution

The Riemann Hypothesis (RH) is a central unsolved problem in mathematics. This research proposes novel approaches to proving the RH by analyzing the distribution of prime numbers and utilizing advanced discrepancy theory. Instead of directly tackling the zeta function, we focus on the fine-grained distribution of primes, intrinsically linked to the RH. This builds upon the work in arXiv:XXXX.XXXXX.

1.1 Discrepancy Theory in Zeta Function Analysis

Our approach uses discrepancy theory to analyze sequences related to prime numbers. We introduce several discrepancy measures:

• Standard Discrepancy: D_N(S) ≤ 1/(H+1) + Σ_h=1^H (1/h)(1/N) |Σ_n=1^N e_h(v_n)|
• Logarithmically Weighted Discrepancy: D_N^log(S) ≤ (3/2) (2/(H+1) + Σ_h=1^H (1/h) (1/Σ_n=1^N (1/n)) |Σ_n=1^N (e_h(v_n)/n)|
• Prime-Weighted Discrepancy: D_N^{log log}(S) ≤ (3/2) (2/(H+1) + Σ_h=1^H (1/h) (1/Σ_n=1^N (1/p_n)) |Σ_n=1^N (e_h(v_n)/p_n)|

where e_h(v_n) = exp(2iπhv_n) and S = (v_n)_n≥1 is a sequence in [0, 1).

1.2 Novel Research Approaches

We propose two approaches combining elements from arXiv:XXXX.XXXXX with existing RH research.

Approach 1: Connecting Discrepancy to Zeta Zeros

Mathematical Foundation: Define sequences S₁ = (t/(2π)log n mod 1)_n≥1 and S₂ = (t/(2π)log p_n mod 1)_n≥1. We derive bounds for D_N^log(S₁) and D_N^{log log}(S₂) and investigate their relationship to the distribution of zeta zeros. Uniformly small discrepancies (or specific behavior) for σ > 1/2 may constrain zero locations.

Approach 2: Exponential Sum Estimates and the Zeta Function

Mathematical Foundation: The sums Σ_n=1^N n^-1+it and Σ_n=1^N p_n^-1+it appear in the discrepancy definitions. We leverage exponential sum estimates to bound these sums, directly connecting them to the zeta function's behavior near the critical line.

2. Research Agenda

Our research agenda includes:

• Conjecture 1: Tight bounds on D_N^log(S₁) and D_N^{log log}(S₂) reveal a relationship to the distribution of zeta zeros.
• Conjecture 2: Sharp bounds for Σ_n=1^N n^-1+it and Σ_n=1^N p_n^-1+it constrain the location of zeta zeros.
• Mathematical Tools: Discrepancy theory, exponential sum estimates, analytic number theory.
• Intermediate Results: Bounds on the growth rate of partial sums of Dirichlet series, relationships between prime distribution and zeta function behavior.

By focusing on the distribution of prime numbers and using discrepancy theory, we believe a new path towards solving the Riemann Hypothesis has been opened.