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

We present a complete proof of the Riemann Hypothesis. The proof proceeds in three stages. Stage 1 develops a novel sieve framework detecting prime pairs in intervals (p, q2 ) for consecutive primes p < q. A sieve-primality equivalence theorem circumvents the parity barrier. Using CRT-based residue class constructions, resonance-breaking double inclusion-exclusion, and symmetric polynomial tail estimates, we prove the Polignac Conjecture and the unconditional bound pn+1 − pn = O(p 1/2+ε n ). Stage 2 embeds the sieve error into GL(2) spectral theory. Dirichlet character expansion introduces Gauss sums connected to Eisenstein series. The crucial technique of complete k-summation converts exponential sums to Ramanujan sums cd(n), whose divisor representation cd(n) = P e|gcd(d,n) eµ(d/e) enables absolute convergence. After rigorous treatment of continuous spectrum, discrete spectrum truncation, and contour integral remainders, we establish ap = P ρ p 2β−2/(ρ(ρ + 1))e 2iγ log p + O(p −2+o(1)), breaking the square-root barrier. Stage 3 constructs FP (ω) = 1 π(P) P p≤P papp −iω and proves: if RH holds, FP (ω) is uniformly bounded; if β0 > 1/2 exists, |FP (2γ0)| → ∞ at rate P 2β0−1/ log P. Contradiction with the unconditional bound |ap| ≤ Cp−1+ε completes the proof. Comments and feedback are welcome. Please feel free to contact me via wuhaizhu0512@163.com