Research Pathways Towards a Proof of the Riemann Hypothesis

Introduction

This research agenda investigates the Riemann Hypothesis, drawing upon the mathematical frameworks presented in arXiv:2312.07095, and integrating them with existing research. The primary focus is on constructing explicit pathways toward a potential proof, emphasizing rigorous mathematical analysis combined with innovative conjectures and computational explorations.

Key Mathematical Frameworks

1. Euler Product and Trigonometric Transformations

The core lies in relating the Euler product formula to trigonometric functions involving prime numbers:

• ζ(s)ζ(s) = ∏_{p ∈ P} 1/((1-e^-xn(p)cos(yln(p)))² + (e^-xn(p)sin(yln(p)))²)

A proposed lemma suggests that if the non-trivial zeros are symmetric about the critical line, then these trigonometric transformations might preserve critical zero properties.

2. Holomorphic Characterization

This framework explores the holomorphic properties of functions like:

• ∏_{p ∈ P} 1/(1-B/p²)

A proposed theorem aims to establish conditions where this product remains non-vanishing and holomorphic near the critical line.

3. Functional Equation Connection

This framework utilizes the known functional equation:

• ζ(s) = 2^sπ^s-1sin(πs/2)Γ(1-s)ζ(1-s)

and connects it to product transformations involving e^-zn(p).

Novel Combined Approaches

1. Product-Symmetry Analysis

This approach investigates the behavior of ∏_{p ∈ P} 1/((1+e^-xn(p))² + (e^-xn(p))²) as s approaches potential zeros, focusing on symmetry properties around the critical line and developing bounds within the critical strip.

2. Prime-Modulated Trigonometric Inequalities

This approach develops inequalities based on the Euler product relationship and explores their implications for the zeros of ζ(s) using Fourier analysis to constrain zero locations.

Tangential Connections and Conjectures

1. Digital Convexity in Prime Distribution

This explores whether digitally convex polyominoes can model the distribution of zeros or primes, conjecturing that their boundary conditions might mimic the statistical behavior of ζ(s) zeros. Computational simulations could analyze perimeter-to-area ratios compared to fluctuations in prime/zero density.

Detailed Research Agenda

The agenda focuses on proving conjectures related to the Euler product's symmetry and holomorphic function properties. It requires tools from complex analysis and analytic number theory, with intermediate results involving bounds on prime product growth. The logical sequence involves theorems establishing these bounds and their connection to zeros, potentially leading to a proof by contradiction. Simplified cases would analyze the inequality for restricted prime sets or specific x and y values.

Conclusion

This research plan offers a structured approach to investigating the Riemann Hypothesis, leveraging the insights from arXiv:2312.07095 and combining them with existing mathematical frameworks. The focus on rigorous analysis, explicit conjectures, and computational experimentation aims to create a concrete path toward a potential proof.