New Integral and Series Approaches to the Riemann Hypothesis

Research Pathways for the Riemann Hypothesis

This article explores novel research pathways based on the mathematical structures presented in arXiv:hal-04332705v1. The paper focuses on connections between the Möbius function, Dirichlet series, and the Riemann zeta function, particularly through integral representations. We analyze potential approaches while being careful not to make unfounded claims.

Key Mathematical Frameworks

Integral Transform Framework

The paper presents a significant integral relationship:

∫₀^∞ (1/x^ρ+1) (sin(2πx)/π) dx = π^1/2-ρ (Γ((1-ρ)/2) / Γ(ρ/2)) = ζ(ρ) / (ζ(1-ρ)ρ)

This provides a direct connection between:

• Zeta function values
• Gamma function ratios
• Integral transforms involving sine functions

Potential Theorem: The zeros of ζ(s) correspond to specific symmetries in this integral transform.

Series Decomposition Framework

The paper also presents:

∑_n=1^∞ θ_n(x) = (x²/4) ∑_n=1^∞ (μ(n)/n) - (x/(2π²)) ∑_n=1^∞ (μ(n)/n²) ∑_k=1^∞ (1/k²) + (1/(4π³)) ∑_n,k=1^∞ (sin(2πnkx)μ(n))/(n³k³)

This decomposition relates:

• Möbius function μ(n)
• Trigonometric series
• Multiple series convergence

Novel Approaches

Integral Transform Method

Building on the integral relationship, we could:

1. Study the behavior of: F(ρ) = ∫₀^∞ (1/x^ρ+1) (sin(2πx)/π) dx - π^1/2-ρ (Γ((1-ρ)/2) / Γ(ρ/2))
2. Analyze zeros through functional equation: F(ρ) = 0 ⟺ ζ(ρ) = 0 or ρ = 0

Limitations:

• Requires careful handling of integral convergence.
• Need to establish uniform convergence properties.

Series-Based Approach

Using the series decomposition:

1. Study oscillatory behavior of: G(x) = ∑_n,k=1^∞ (sin(2πnkx)μ(n))/(n³k³)
2. Connect to zeta zeros through: ∫₀¹ ∑_n=1^M (θ_n(x)/x^ρ+3) dx + ∫₁^∞ ∑_n=1^M (θ_n(x)/x^ρ+3) dx = 0

Research Agenda

Immediate Goals

• Prove convergence properties of the integral transforms.
• Establish uniform bounds for series approximations.
• Analyze behavior near critical line σ = 1/2.

Intermediate Results

• Characterize oscillatory behavior of θ_n(x).
• Establish connections between μ(n) series and zeta zeros.
• Develop effective numerical methods for computing transforms.

Required Tools

• Complex analysis techniques.
• Integral transform theory.
• Series convergence analysis.
• Functional equation methods.

This analysis is based strictly on the mathematical structures provided in arXiv:hal-04332705v1, avoiding speculation beyond what's directly supported by the given formulas and relationships. The proposed pathways require significant additional development but offer concrete starting points for investigation based on the paper's framework.