A Complete Proof of the Riemann Hypothesis via Spectral Analysis of Sieve Errors and Ramanujan Summation

Key Points

• This research aims to provide a complete proof of the Riemann Hypothesis through a structured framework involving sieve analysis and spectral theory.
• Develops a sieve framework detecting prime pairs and proves the Polignac Conjecture.
• Embeds sieve errors into GL(2) spectral theory and employs techniques like complete k-summation.
• Constructs FP(ω) and establishes bounds under the assumption of the Riemann Hypothesis.
• Establishes that if RH holds, FP(ω) is uniformly bounded.
• Demonstrates that under certain conditions, FP(2γ0) approaches infinity at a specific rate.
• Confirms the unconditional bound |ap| ≤ Cp−1+ε, leading to the contradiction necessary to complete the proof.